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Introduction to the Practice of Statistics

David S. Moore; George P. McCabe; Bruce A. Craig

Chapter 7

Inference for Distributions - all with Video Answers

Educators


Chapter Questions

03:12

Problem 1

You randomly choose 16 unfurnished one-bedroom apartments from a large number of advertisements in your local newspaper. You calculate that their mean monthly rent is $\$ 600$ and their standard deviation is $\$ 55$.
(a) What is the standard error of the mean?
(b) What are the degrees of freedom for a one-sample $t$ statistic?

Nick Johnson
Nick Johnson
Numerade Educator
02:04

Problem 2

Refer to the previous exercise. You plan to construct a confidence interval for the average monthly rent of unfurnished one-bedroom apartments in your area. If you were to use $90 \%$ confidence, rather than $95 \%$ confidence, would the margin of error be larger or smaller? Does your answer depend on sample size? Explain your answer.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
03:01

Problem 3

Refer to Exercise $7.1$ (page 420). Construct a $95 \%$ confidence interval for the mean monthly rent of all advertised one-bedroom apartments.

Jon Southam
Jon Southam
Numerade Educator
01:30

Problem 4

What critical value $t^*$ from Table D should be used to construct
(a) a $95 \%$ confidence interval when $n=12$ ?
(b) a $99 \%$ confidence interval when $n=38$ ?
(c) a $90 \%$ confidence interval when $n=81$ ?

James Kiss
James Kiss
Numerade Educator
01:44

Problem 5

A test of a null hypothesis versus a two-sided alternative gives $t=2.22$.
(a) The sample size is 18 . Is the test result significant at the $5 \%$ level? Explain how you obtained your answer.
(b) The sample size is 9 . Is the test result significant at the $5 \%$ level? Explain how you obtained your answer.
(c) Sketch the two $t$ distributions to illustrate your answers.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:58

Problem 6

Refer to Exercise $7.1$ (page 420). Does this SRS give good reason to believe that the mean rent of all advertised one-bedroom apartments is greater than $\$ 550$ ? State the hypotheses, find the $t$ statistic and its $P$ value, and state your conclusion.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:32

Problem 7

In Example $7.1$ (page 421) we calculated the $95 \%$ confidence interval for the U.S. college student average of hours per month spent watching videos on a cell phone. Use software to compute this interval and verify that you obtain the same interval.

Lucas Finney
Lucas Finney
Numerade Educator
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Problem 8

Consider the following study to compare two popular energy drinks. For each subject, a coin was flipped to determine which drink to rate first. Each drink was rated on a 0 to 100 scale, with 100 being the highest rating.
Is there a difference in preference? State appropriate hypotheses and carry out a matched pairs $t$ test for these data.

Emily Himsel
Emily Himsel
Numerade Educator
02:42

Problem 9

Refer to the previous exercise. For the company producing Drink A, the real question is how much difference there is between the two preferences. Use the data in Exercise $7.8$ to give a $95 \%$ confidence interval for the difference in preference between Drink A and Drink B.

Shu Naito
Shu Naito
Numerade Educator
01:56

Problem 10

Consider the data from Exercise $1.47$ (page 32) but with Suriname removed. Would you be comfortable applying the $t$ procedures in this case? Explain your answer.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
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Problem 11

Consider the data on StubHub! ticket prices presented in Figure $1.31$ (page 71). Would you be comfortable applying the $t$ procedures in this case? In explaining your answer, recall that these $t$ procedures focus on the mean $\mu$.

Victor Salazar
Victor Salazar
Numerade Educator
02:31

Problem 12

If you were to repeat the power calculation in Example $7.9$ for a value of $\mu$ that is greater than 1 , would you expect the power to be higher or lower than $89 \%$ ? Why?

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:21

Problem 13

Verify your answer to the previous question by doing the calculation for the alternative $\mu$

Ajay Singhal
Ajay Singhal
Numerade Educator
01:55

Problem 14

If you were to repeat the power calculation in Example $7.9$ using $n=25$ instead of $n=20$, would you expect the power to be higher or lower than $89 \%$ ? Why?

M S
M S
Numerade Educator
02:30

Problem 15

Verify your answer to the previous question by doing the calculation for the alternative $\mu=1$ and $n=25$.

Manish Jain
Manish Jain
Numerade Educator
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Problem 16

Exercise $7.8$ (page 432) gives data on the appeal of two popular energy drinks. Is there evidence that the medians are different? State the hypotheses, carry out the sign test, and report your conclusion.

Victor Salazar
Victor Salazar
Numerade Educator
01:30

Problem 17

What critical value $t^*$ from Table D should be used to calculate the margin of error for a confidence interval for the mean of the population in each of the following situations?
(a) A $95 \%$ confidence interval based on $n=12$ observations.
(b) A $95 \%$ confidence interval from an SRS of 21 observations.
(c) A $90 \%$ confidence interval from a sample of size 21 .
(d) These cases illustrate how the size of the margin of error depends upon the confidence level and the sample size. Summarize these relationships.

James Kiss
James Kiss
Numerade Educator
01:26

Problem 18

Assume a sample size of $n=16$. Draw a picture of the distribution of the $t$ statistic under the null hypothesis. Use Table D and your picture to illustrate the values of the test statistic that would lead to rejection of the null hypothesis at the $5 \%$ level for a two-sided alternative.

Tyler Moulton
Tyler Moulton
Numerade Educator
01:22

Problem 19

Repeat the previous execise for the two situations where the alternative is one-sided.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:25

Problem 20

Computer software reports $\mathrm{x}^{-}=11.2$ and $P=0.068$ for a $t$ test of $H_0: \mu=0$ versus $H_a: \mu \neq 0$. Based on prior knowledge, you justified testing the alternative $H_a: \mu>0$ What is the $P$-value for your significance test?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:25

Problem 21

Suppose that computer software reports $\mathrm{x}^{-}=-11.2$ and $P=0.068$ for a $t$ test of $H_0: \mu=0$ versus $H_a: \mu \neq 0$. Would this change your $P$-value for the alternative hypothesis in the previous execise? Use a sketch of the distribution of the test statistic under the null hypothesis to illustrate and explain your answer.

Tyler Moulton
Tyler Moulton
Numerade Educator
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Problem 22

The one-sample $t$ statistic for testing
$$
\begin{aligned}
&H_0: \mu=8 \\
&H_a:>=8
\end{aligned}
$$
from a sample of $n=16$ observations has the value $t=2.15$
(a) What are the degrees of freedom for this statistic?
(b) Give the two critical values $t^*$ from Table $\mathrm{D}$ that bracket $t$.
(c) Between what two values does the $P$-value of the test fall?
(d) Is the value $t=2.15$ significant at the $5 \%$ level? Is it significant at the $1 \%$ level?
(e) If you have software available, find the exact $P$-value.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
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Problem 23

The one-sample $t$ statistic for testing
$$
\begin{aligned}
&H_0: \mu=40 \\
&H_a: \mu \neq 40
\end{aligned}
$$
from a sample of $n=27$ observations has the value $t=2.01$
(a) What are the degrees of freedom for $t$ ?
(b) Locate the two critical values $t^*$ from Table D that bracket $t$.
(c) Between what two values does the $P$-value of the test fall?
(d) Is the value $t=2.01$ statistically significant at the $5 \%$ level? At the $1 \%$ level?
(e) If you have software available, find the exact $P$-value.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
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Problem 24

The one-sample $t$ statistic for testing
$$
\begin{aligned}
&H_0: \mu=20 \\
&H_a:<=20
\end{aligned}
$$
based on $n=11$ observations has the value $t=-1.85$.
(a) What are the degrees of freedom for this statistic?
(b) Between what two values does the $P$-value of the test fall?
(c) If you have software available, find the exact $P$-value.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
01:09

Problem 25

Most software gives $P$-values for two-sided alternatives. Explain why you cannot always divide these $P$ values by 2 to obtain $P$-values for one-sided alternatives.

Nick Johnson
Nick Johnson
Numerade Educator
03:47

Problem 26

Facebook recently examined all active Facebook users (more than $10 \%$ of the global population) and determined that the average user has 190 friends. This distribution takes only integer values, so it is certainly not Normal. It is also highly skewed to the right, with a median of 100 friends. ${ }^8$ Consider the following SRS of $n=30$ Facebook users from your large university.
$$
\begin{array}{rrrrrrrrrr}
\hline 594 & 60 & 417 & 120 & 132 & 176 & 516 & 319 & 734 & 8 \\
31 & 325 & 52 & 63 & 537 & 27 & 368 & 11 & 12 & 190 \\
85 & 165 & 288 & 65 & 57 & 81 & 257 & 24 & 297 & 148 \\
\hline
\end{array}
$$(a) Are these data also heavily skewed? Use graphical methods to examine the distribution. Write a short summary of your findings.
(b) Do you think it is appropriate to use the $t$ methods of this section to compute a $95 \%$ confidence interval for the mean number of friends that Facebook users at your large university have? Explain why or why not.
(c) Compute the sample mean and standard deviation, the standard error of the mean, and the margin of error for $95 \%$ confidence.
(d) Report the $95 \%$ confidence interval for $\mu$, the average number of friends for Facebook users at your large university.

Nick Johnson
Nick Johnson
Numerade Educator
06:00

Problem 27

In February 2013, two California residents filed a class-action lawsuit against Anheuser-Busch, alleging the company was watering down beers to boost profits. 9 They argued that because water was being added, the true alcohol content of the beer by volume is less than the advertised amount. For example, they alleged that Budweiser beer has an alcohol content by volume of $4.7 \%$ instead of the stated $5 \%$. Several media outlets picked up on this suit and hired independent labs to test samples of Budweiser beer and find the alcohol content. Below is a summary of these tests, each done on a single can. BU.1. BU (a) Even though we have a very small sample, test the null hypothesis that the alcohol content is $4.7 \%$ by volume. Do the data provide evidence against the claim of the two residents?
(b) Construct a $95 \%$ confidence interval for the true alcohol content in Budweiser.
(c) U.S. government standards require that the true alcohol content in all cans and bottles be within $\pm 0.3 \%$ of the advertised level. Do these tests provide strong evidence that this is the case for Budweiser beer? Explain your answer.

Jon Southam
Jon Southam
Numerade Educator
03:16

Problem 28

The Nielsen Company reported that U.S. residents aged 18 to 24 years spend an average of $35.5$ hours per month using the Internet on a computer. ${ }^{10}$ You think this is quite low compared with the amount of time that students at your university spend using the Internet on a computer, and you decide to do a survey to verify this. You collect an SRS of $n=50$ students and obtain $\mathrm{x}^{-}=40.1$ hours with $s=28.4$ hours.
(a) Report the $95 \%$ confidence interval for $\mu$ the average number of hours per month that students at your university use the Internet on a computer.
(b) Use this interval to test whether the average time for students at your university is different from the average reported by Nielsen. Use the $5 \%$ significance level. Summarize your results.

Nick Johnson
Nick Johnson
Numerade Educator
02:15

Problem 29

Many believe that an uncivil environment has a negative effect on people. A pair of researchers performed a series of experiments to test whether witnessing rudeness and disrespect affects task performance. 11 In one study, 34 participants met in small groups and witnessed the group organizer being rude to a "participant" who showed up late for the group meeting. After the exchange, each participant performed an individual brainstorming task in which he or she was asked to produce as many uses for a brick as possible in 5 minutes. The mean number of uses was $7.88$ with a standard deviation of $2.35$.
(a) Suppose that prior research has shown that the average number of uses a person can produce in 5 minutes under normal conditions is 10 . Given that the researchers hypothesize that witnessing this rudeness will decrease performance, state the appropriate null and alternative hypotheses.
(b) Carry out the significance test using a significance level of $0.05$. Give the $P$-value and state your conclusion.

Jameson Kuper
Jameson Kuper
Numerade Educator
04:43

Problem 30

Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the miles per gallon were recorded each time the gas tank was filled, and the computer was then reset. $^{12}$ Here are the mpg values for a random sample of 20 of these records: MPG
$$
\begin{array}{llllllllll}
\hline 41.5 & 50.7 & 36.6 & 37.3 & 34.2 & 45.0 & 48.0 & 43.2 & 47.7 & 42.2 \\
43.2 & 44.6 & 48.4 & 46.4 & 46.8 & 39.2 & 37.3 & 43.5 & 44.3 & 43.3 \\
\hline
\end{array}
$$
(a) Describe the distribution using graphical methods. Is it appropriate to analyze these data using methods based on Normal distributions? Explain why or why not.
(b) Find the mean, standard deviation, standard error, and margin of error for $95 \%$ confidence.
(c) Report the $95 \%$ confidence interval for $\mu$, the mean miles per gallon for this vehicle based on these

Samuel Goyette
Samuel Goyette
Numerade Educator
01:52

Problem 31

A study of 584 longleaf pine trees in the Wade Tract in Thomas County, Georgia, is described in Example $6.1$ (page 352). For each tree in the tract, the researchers measured the diameter at breast height (DBH). This is the diameter of the tree at a height of $4.5$ feet, and the units are centimeters $(\mathrm{cm})$. Only trees with DBH greater than $1.5 \mathrm{~cm}$ were sampled. Here are the diameters of a random sample of 40 of these trees:
PINES
$$
\begin{array}{rrrrrrrrrr}
\hline 10.5 & 13.3 & 26.0 & 18.3 & 52.2 & 9.2 & 26.1 & 17.6 & 40.5 & 31.8 \\
47.2 & 11.4 & 2.7 & 69.3 & 44.4 & 16.9 & 35.7 & 5.4 & 44.2 & 2.2 \\
4.3 & 7.8 & 38.1 & 2.2 & 11.4 & 51.5 & 4.9 & 39.7 & 32.6 & 51.8 \\
43.6 & 2.3 & 44.6 & 31.5 & 40.3 & 22.3 & 43.3 & 37.5 & 29.1 & 27.9 \\
\hline
\end{array}
$$
(a) Use a histogram or stemplot and a boxplot to examine the distribution of DBHs. Include a Normal quantile plot if you have the necessary software. Write a careful description of the distribution.
(b) Is it appropriate to use the methods of this section to find a $95 \%$ confidence interval for the mean DBH of all trees in the Wade Tract? Explain why or why not.
(c) Report the mean with the margin of error and the confidence interval. Write a short summary describing the meaning of the confidence interval.
(d) Do you think these results would apply to other similar trees in the same area? Give reasons for your answer.

KS
Kathleen Snyder
Numerade Educator
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Problem 32

Recall Exercise $6.72$ (page 393). For one part of the study, $n=114$ male athletes from eight Canadian sports centers were surveyed. Their average caloric intake was $3077.0$ kilocalories per day ( $\mathrm{kcal} / \mathrm{d}$ ) with a standard deviation of 987.0. The recommended amount is 3421.7. Is there evidence that Canadian highperformance male athletes are deficient in their caloric intake?
(a) State the appropriate $H_0$ and $H_a$ to test this.
(b) Carry out the test, give the $P$-value, and state your conclusion.
(c) Construct a $95 \%$ confidence interval for the average deficiency in caloric intake.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
04:21

Problem 33

We often feel that the return trip from a destination takes less time than the trip to the destination even though the distance traveled is usually identical. To better understand this effect, a group of researchers ran a series of experiments. ${ }^{13}$ In one experiment, they surveyed 69 participants who had just returned from a day trip by bus. Each was asked to rate how long the return trip had taken, compared with the initial trip, on an 11-point scale from $-5=$ a lot shorter to $5=a$ lot longer. The sample mean was $-0.55$ and the sample standard deviation was $2.16$.
(a) These data are integer values. Do you think we can still use the $t$-based methods of this section? Explain your answer.
(b) Is there evidence that the mean rating is different from zero? Carry out the significance test using $\alpha=$ 0.05 \text { and summarize the results. }

Lucas Finney
Lucas Finney
Numerade Educator
02:41

Problem 34

In a study of parents who have children with attention-deficit/hyperactivity disorder (ADHD), parents were asked to rate their overall stress level using the Parental Stress Scale (PSS). ${ }^{14}$ This scale has 18 items that contain statements regarding both positive and negative aspects of parenthood. Respondents are asked to rate their agreement with each statement using a 5 -point scale ( $1=$ strongly disagree to $5=$ strongly agree). The scores are summed such that a higher score indicates greater stress. The mean rating for the 50 parents in the study was reported as $52.98$ with a standard deviation of $10.34$.
(a) Do you think that these data are approximately Normally distributed? Explain why or why not.
(b) Is it appropriate to use the methods of this section to compute a $90 \%$ confidence interval? Explain why or why not.
(c) Find the $90 \%$ margin of error and the corresponding confidence interval. Write a sentence explaining the interval and the meaning of the $90 \%$ confidence level.
(d) To recruit parents for the study, the researchers visited a psychiatric outpatient service in Rohtak, India, and selected 50 consecutive families who met the inclusion and exclusion criteria. To what extent do you think the results can be generalized to all parents with children who have ADHD in India or in other locations around the world?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 35

Refer to the previous execise. The researchers considered a score greater than 45 to represent extreme stress. Is there evidence that the average stress level for the parents in this study is above this level? Perform a test of significance using $\alpha=0.10$ and summarize your results.

Victor Salazar
Victor Salazar
Numerade Educator
02:35

Problem 36

If we increase our food intake, we generally gain weight. Nutrition scientists can calculate the amount of weight gain that would be associated with a given increase in calories. In one study, 16 nonobese adults, aged 25 to 36 years, were fed 1000 calories per day in excess of the calories needed to maintain a stable body weight. The subjects maintained this diet for 8 weeks, so they consumed a total of 56,000 extra calories. ${ }^{15}$ According to theory, 3500 extra calories will translate into a weight gain of 1 pound. Therefore, we expect each of these subjects to gain $56,000 / 3500=16$ pounds $(\mathrm{lb})$. Here are the weights before and after the 8-week period, expressed in kilograms $(\mathrm{kg})$ : 1., WTGAIN
$$
\begin{array}{l|rrrrrrrr}
\hline \text { Subject } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\text { Weight before } & 55.7 & 54.9 & 59.6 & 62.3 & 74.2 & 75.6 & 70.7 & 53.3 \\
\text { Weight after } & 61.7 & 58.8 & 66.0 & 66.2 & 79.0 & 82.3 & 74.3 & 59.3 \\
\hline \text { Subject } & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\text { Weight before } & 73.3 & 63.4 & 68.1 & 73.7 & 91.7 & 55.9 & 61.7 & 57.8 \\
\text { Weight after } & 79.1 & 66.0 & 73.4 & 76.9 & 93.1 & 63.0 & 68.2 & 60.3 \\
\hline
\end{array}
$$
(a) For each subject, subtract the weight before from the weight after to determine the weight change.
(b) Find the mean and the standard deviation for the weight change.
(c) Calculate the standard error and the margin of error for $95 \%$ confidence. Report the $95 \%$ confidence
interval for weight change in a sentence that explains the meaning of the $95 \%$.
(d) Convert the mean weight gain in kilograms to mean weight gain in pounds. Because there are $2.2 \mathrm{~kg}$ per pound, multiply the value in kilograms by $2.2$ to obtain pounds. Do the same for the standard deviation and the confidence interval.
(e) Test the null hypothesis that the mean weight gain is $16 \mathrm{lb}$. Be sure to specify the null and alternative hypotheses, the test statistic with degrees of freedom, and the $P$-value. What do you conclude?
(f) Write a short paragraph explaining your results.

Nick Johnson
Nick Johnson
Numerade Educator
11:12

Problem 37

Nonexecise activity thermogenesis (NEAT) provides a partial explanation for the results you found in the previous analysis. NEAT is energy burned by fidgeting, maintenance of posture, spontaneous muscle contraction, and other activities of daily living. In the study of the previous execise, the 16 subjects increased their NEAT by 328 calories per day, on average, in response to the additional food intake. The standard deviation was 256.
(a) Test the null hypothesis that there was no change in NEAT versus the two-sided alternative. Summarize the results of the test and give your conclusion.
(b) Find a $95 \%$ confidence interval for the change in NEAT. Discuss the additional information provided by the confidence interval that is not evident from the results of the significance test.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:43

Problem 38

Insurance adjusters are concerned about the high estimates they are receiving from Jocko's Garage. To see if the estimates are unreasonably high, each of 10 damaged cars was taken to Jocko's and to another garage and the estimates (in dollars) were recorded. Here are the results:
JOCKO
$$
\begin{array}{l|rrrrr}
\hline \text { Car } & 1 & 2 & 3 & 4 & 5 \\
\text { Jocko's } & 1410 & 1550 & 1250 & 1300 & 900 \\
\text { Other } & 1250 & 1300 & 1250 & 1200 & 950 \\
\text { Car } & 6 & 7 & 8 & 9 & 10 \\
\text { Jocko's } & 1520 & 1750 & 3600 & 2250 & 2840 \\
\text { Other } & 1575 & 1600 & 3380 & 2125 & 2600 \\
\hline
\end{array}
$$
(a) For each car, subtract the estimate of the other garage from Jocko's estimate. Find the mean and the standard deviation for this difference.
(b) Test the null hypothesis that there is no difference between the estimates of the two garages. Be sure to specify the null and alternative hypotheses, the test statistic with degrees of freedom, and the $P$-value. What do you conclude using the $0.05$ significance level?
(c) Construct a $95 \%$ confidence interval for the difference in estimates.
(d) The insurance company is considering seeking repayment from 1000 claims filed with Jocko's last year. Using your answer to part (c), what repayment would you recommend the insurance company seek? Explain your answer.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:31

Problem 39

Refer to Exercise 7.30. In addition to the computer calculating miles per gallon, the driver also recorded the miles per gallon by dividing the miles driven by the number of gallons at fill-up. The driver wants to determine if these calculations are different. MPGDIFF
$$
\begin{aligned}
&\begin{array}{l|cccccccccc}
\hline \text { Fill-up } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10
\end{array}\\
&\begin{array}{l|rrrrrrrrrr}
\text { Driver } & 36.5 & 44.2 & 37.2 & 35.6 & 30.5 & 40.5 & 40.0 & 41.0 & 42.8 & 39.2 \\
\hline \text { Fill-up } & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\
\text { Computer } & 43.2 & 44.6 & 48.4 & 46.4 & 46.8 & 39.2 & 37.3 & 43.5 & 44.3 & 43.3 \\
\text { Driver } & 38.8 & 44.5 & 45.4 & 45.3 & 45.7 & 34.2 & 35.2 & 39.8 & 44.9 & 47.5 \\
\hline
\end{array}
\end{aligned}
$$
(a) State the appropriate $H_0$ and $H_a$
(b) Carry out the test using a significance level of $0.05$. Give the $P$-value, and then interpret the result.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
06:40

Problem 40

A guitar supply company must maintain strict oversight on the number of picks they package for sale to customers. Their current advertisement specifies between 900 and 1000 picks in every bag. An SRS of 36 one-pound bags of picks was collected as part of a quality improvement effort within the company. The number of picks in each bag is shown in the following table. $1.1$ PICKS
(a) Create (i) a histogram or a stemplot, (ii) a boxplot, and (iii) a Normal quantile plot of these counts. Write a careful description of the distribution. Make sure to note any outliers, and comment on the skewness and Normality of the data.
(b) Based on your observations in part (a), is it appropriate to analyze these data using the $t$ procedures? Briefly explain your response.
(c) Find the mean, the standard deviation, and the standard error of the mean for this sample.
(d) Calculate the $90 \%$ confidence interval for the mean number of picks in a one-pound bag.

James Kiss
James Kiss
Numerade Educator
02:01

Problem 41

Refer to the previous execise.
(a) Do these data provide evidence that the average number of picks in a one-pound bag is greater than 925 ? Using a significance level of $5 \%$, state your hypotheses, the $P$-value, and your conclusions.
(b) Do these data provide evidence that the average number of picks in a one-pound bag is greater than 935? Using a significance level of $5 \%$, state your hypotheses, the $P$-value, and your conclusion.
(c) Explain the relationship between your conclusions in parts (a) and (b) and the $90 \%$ confidence interval calculated in the previous problem.

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 42

Many organizations are doing surveys to determine the satisfaction of their customers. Attitudes toward various aspects of campus life were the subject of one such study conducted at Purdue University. Each item was rated on a 1 to 5 scale, with 5 being the highest rating. The average response of 2368 first-year students to "Feeling welcomed at Purdue" was $3.92$ with a standard deviation of $1.02$. Assuming that the respondents are an SRS, give a $90 \%$ confidence interval for the mean of all first-year students.

Alexandra Doss
Alexandra Doss
Numerade Educator
01:27

Problem 43

Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab, and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators. ${ }^{16}$ The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data:
(a) Take the difference between the TBBMC recorded for Operator 1 and the TBBMC for Operator 2 .
Describe the distribution of these differences. Is it appropriate to analyze these data using the $t$ methods?
Explain why or why not.
(b) Use a significance test to examine the null hypothesis that the two operators have the same mean. Be sure to give the test statistic with its degrees of freedom, the $P$-value, and your conclusion.
(c) The sample here is rather small, so we may not have much power to detect differences of interest. Use a $95 \%$ confidence interval to provide a range of differences that are compatible with these data.
(d) The eight subjects used for this comparison were not a random sample. In fact, they were friends of the researchers whose ages and weights were similar to these of the types of people who would be measured with this DXA machine. Comment on the appropriateness of this procedure for selecting a sample, and discuss any consequences regarding the interpretation of the significance-testing and confidence interval results.

Tyler Moulton
Tyler Moulton
Numerade Educator
03:12

Problem 44

Refer to the previous execise. TBBMC measures the total amount of mineral in the bones. Another important variable is total body bone mineral density (TBBMD). This variable is calculated by dividing TBBMC by the area corresponding to bone in the DXA scan. The units are grams per squared centimeter $\left(\mathrm{g} / \mathrm{cm}^2\right)$. Here are the TBBMD values for the same subjects:
Analyze these data using the questions in the previous execise as a guide.

John Long
John Long
Numerade Educator
03:09

Problem 45

The National Endowment for the Humanities sponsors summer institutes to improve the skills of high school teachers of foreign languages. One such institute hosted 20 French teachers for 4 weeks. At the beginning of the period, the teachers were given the Modern Language Association's listening test of understanding of spoken French. After 4 weeks of immersion in French in and out of class, the listening test was given again. (The actual French spoken in the two tests was different, so that simply taking the first test should not improve the score on the second test.) The maximum possible score on the test is $36.17$
Here are the data: SUMLANG
$$
\begin{array}{cccc|cccc}
\hline \text { Teacher } & \text { Pretest } & \text { Posttest } & \text { Gain } & \text { Teacher } & \text { Pretest } & \text { Posttest } & \text { Gain } \\
\hline 1 & 32 & 34 & 2 & 11 & 30 & 36 & 6 \\
2 & 31 & 31 & 0 & 12 & 20 & 26 & 6 \\
3 & 29 & 35 & 6 & 13 & 24 & 27 & 3 \\
4 & 10 & 16 & 6 & 14 & 24 & 24 & 0 \\
5 & 30 & 33 & 3 & 15 & 31 & 32 & 1 \\
6 & 33 & 36 & 3 & 16 & 30 & 31 & 1 \\
7 & 22 & 24 & 2 & 17 & 15 & 15 & 0 \\
8 & 25 & 28 & 3 & 18 & 32 & 34 & 2 \\
9 & 32 & 26 & -6 & 19 & 23 & 26 & 3 \\
10 & 20 & 26 & 6 & 20 & 23 & 26 & 3 \\
\hline
\end{array}
$$
To analyze these data, we first subtract the pretest score from the posttest score to obtain the improvement for each teacher. These 20 differences form a single sample. They appear in the "Gain" columns. The first teacher, for example, improved from 32 to 34 , so the gain is $34-32=2$
(a) State appropriate null and alternative hypotheses for examining the question of whether or not the course improves French spoken-language skills.
(b) Describe the gain data. Use numerical and graphical summaries.
(c) Perform the significance test. Give the test statistic, the degrees of freedom, and the $P$-value. Summarize your conclusion.
(d) Give a $95 \%$ confidence interval for the mean improvement.

AG
Ankit Gupta
Numerade Educator
02:22

Problem 46

Refer to the lengths of calls to a customer service center in Table $1.2$ (page 19). Give graphical and numerical summaries for these data. Compute a $95 \%$ confidence interval for the mean call length.
Comment on the validity of your interval. CALLS80

Lucas Finney
Lucas Finney
Numerade Educator
01:55

Problem 47

The differences in the repair estimates in Exercise $7.38$ can also be analyzed using a sign test. Set up the appropriate null and altemative hypotheses, carry out the test, and summarize the results. How do these results compare with those that you obtained in Exercise 7.38?

Raymond Matshanda
Raymond Matshanda
Numerade Educator
01:55

Problem 48

The differences in the TBBMC measures in Exercise $7.43$ can also be analyzed using a sign test. Set up the appropriate null and alternative hypotheses, carry out the test, and summarize the results. How do these results compare with those that you obtained in Exercise $7.43 ?$

Raymond Matshanda
Raymond Matshanda
Numerade Educator
01:59

Problem 49

TBBMD values for the same subjects that you studied in the previous execise are given in Exercise 7.44. Answer the questions given in the previous execise for TBBMD. TBBMD

Yingtai Xiao
Yingtai Xiao
Numerade Educator
01:23

Problem 50

Use the sign test to assess whether the summer institute of Exercise $7.45$ improves French listening skills.
State the hypotheses, give the $P$-value using the binomial table (Table $\mathrm{C}$ ), and report your conclusion. SUMLANG

Tyler Moulton
Tyler Moulton
Numerade Educator
01:37

Problem 51

Use the sign test to assess whether the computer calculates a higher mpg than the driver in Exercise 7.39.
State the hypotheses, give the $P$-value using the binomial table (Table $\mathrm{C}$ ), and report your conclusion.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:36

Problem 52

A manufacturer of electric motors tests insulation at a high temperature $250^{\circ} \mathrm{C}$ ) and records the number of hours until the insulation fails. 18 The data for 5 specimens are
$$
\begin{array}{lllll}
446 & 326 & 372 & 377 & 310
\end{array}
$$
The small sample size makes judgment from the data difficult, but engineering experience suggests that the logarithm of the failure time will have a Normal distribution. Take the logarithms of the 5 observations, and use $t$ procedures to give a $90 \%$ confidence interval for the mean of the log failure time for insulation of this type. II, INSULAT

Adriano Chikande
Adriano Chikande
Numerade Educator
01:00

Problem 53

Suppose that the bone researchers in Exercise $7.43$ wanted to be able to detect an alternative mean difference of $0.002$. Find the power for this alternative for a sample size of 15 . Use the standard deviation that you found in Exercise $7.43$ for these calculations.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:11

Problem 54

You are designing a study to test the null hypothesis that $\mu=0$ versus the alternative that $\mu$ is positive. Assume that $\sigma$ is 15 . Suppose that it would be important to be able to detect the alternative $\mu=2$ Perform power calculations for a variety of sample sizes and determine how large a sample you would need to detect this alternative with power of at least $0.80$.

Amany Waheeb
Amany Waheeb
Numerade Educator
03:11

Problem 55

Consider Example $7.9$ (page 435 ). What is the minimum sample size needed for the power to be greater than $80 \%$ when $\mu=0.75$ ?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:30

Problem 56

Assume that $\mathrm{x}^{-} 1=110, \mathrm{x}^{-} 2=120, s_1=8, s_2=12, n_1=50$, and $n_2=50$. Find a $95 \%$ confidence interval for the difference in the corresponding values of $\mu$ using the second approximation for degrees of freedom. Does this interval include more or fewer values than a $99 \%$ confidence interval would? Explain your answer.

Akhil Choudhary
Akhil Choudhary
Numerade Educator
03:05

Problem 57

Assume that $\mathrm{x}^{-} 1=110, \mathrm{x}^{-} 2=120, s_1=8, s_2=12, n_1=10$, and $n_2=10$. Find a $95 \%$ confidence interval for the difference in the corresponding values of $\mu$ using the second approximation for degrees of freedom. Would you reject the null hypothesis that the population means are equal in favor of the two-sided alternative at significance level $0.05$ ? Explain.

Prashant Bana
Prashant Bana
Numerade Educator
12:13

Problem 58

You want to compare the daily number of hits for two different MySpace page designs that advertise your indie rock band. You assign the next 30 days to either Design A or Design B, 15 days to each.
(a) Would you use a one-sided or a two-sided significance test for this problem? Explain your choice.
(b) If you use Table D to find the critical value, what are the degrees of freedom using the second approximation?
(c) If you perform the significance test using $\alpha=0.05$, how large (positive or negative) must the $t$ statistic be to reject the null hypothesis that the two designs result in the same average number of hits?

R M
R M
Numerade Educator
00:38

Problem 59

Refer to the previous exercise. If the $t$ statistic for comparing the mean hits was $2.18$, what $P$-value would you report? What would you conclude using $\alpha=0.05 ?$

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
02:16

Problem 60

Assume that $s_1=13, s_2=8, n_1=28$, and $n_2=24$. Find the approximate degrees of freedom.

James Kiss
James Kiss
Numerade Educator
03:53

Problem 61

Figure $7.15$ (pages 458-460) gives the outputs from four software packages for comparing the weight loss of two groups with different eating schedules. Some of the software reports both pooled and unpooled analyses. Which outputs give the pooled results? What are the pooled $t$ and its $P$-value?

Beth Stone
Beth Stone
Numerade Educator
00:18

Problem 62

The software outputs in Figure $7.15$ give the same value for the pooled and unpooled $t$ statistics. Do some simple algebra to show that this is always true when the two sample sizes $n_1$ and $n_2$ are the same. In other cases, the two $t$ statistics usually differ.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
03:07

Problem 63

In each of the following situations explain what is wrong and why.
(a) A researcher wants to test $\mathrm{H} 0: \mathrm{x}^{-} 1=\mathrm{x}^{-} 2$ versus the two-sided alternative Ha: $\mathrm{x}^{-} 1 \neq \mathrm{x}^{-} 2$
(b) A study recorded the IQ scores of 100 college freshmen. The scores of the 56 males in the study were compared with the scores of all 100 freshmen using the two-sample methods of this section.
(c) A two-sample $t$ statistic gave a $P$-value of $0.94$. From this we can reject the null hypothesis with $90 \%$ confidence.
(d) A researcher is interested in testing the one-sided alternative $H_a: \mu<\mu_2$. The significance test gave $t=2.15$. Since the $P$-value for the two-sided alternative is $0.036$, he concluded that his $P$-value

Lucas Finney
Lucas Finney
Numerade Educator
02:47

Problem 64

For each of the following, answer the question and give a short explanation of your reasoning.
(a) A $95 \%$ confidence interval for the difference between two means is reported as $(0.8,2.3)$. What can you conclude about the results of a significance test of the null hypothesis that the population means are equal versus the two-sided alternative?
(b) Will larger samples generally give a larger or smaller margin of error for the difference between two sample means?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:27

Problem 65

For each of the following, answer the question and give a short explanation of your reasoning.
(a) A significance test for comparing two means gave $t=-1.97$ with 10 degrees of freedom. Can you reject the null hypothesis that the $\mu$ 's are equal versus the two-sided alternative at the $5 \%$ significance level?
(b) Answer part (a) for the one-sided alternative that the difference between means is negative.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:56

Problem 66

Assume that $\mathrm{x}^{-} 1=100, \mathrm{x}^{-} 2=115, s 1=19, s 2=16, n_1=50$, and $n_2=40$. Find a $95 \%$ confidence interval for the difference between the corresponding values of $\mu$. Does this interval include more or fewer values than a $99 \%$ confidence interval would? Explain your answer.

Lucas Finney
Lucas Finney
Numerade Educator
03:30

Problem 67

Why do we naturally tend to trust some strangers more than others? One group of researchers decided to study the relationship between eye color and trustworthiness. ${ }^{25}$ In their experiment the researchers took photographs of 80 students ( 20 males with brown eyes, 20 males with blue eyes, 20 females with brown eyes, and 20 females with blue eyes), each seated in front of a white background looking directly at the camera with a neutral expression. These photos were cropped so the eyes were horizontal and at the same height in the photo and so the neckline was visible. They then recruited 105 participants to judge the trustworthiness of each student photo. This was done using a 10-point scale, where 1 meant very untrustworthy and 10 very trustworthy. The 80 scores from each participant were then converted to $z$-scores, and the average $z$-score of each photo (across all 105 participants) was used for the analysis. Here is a summary of the results: Can we conclude from these data that brown-eyed students appear more trustworthy compared to their blue-eyed counterparts? Test the hypothesis that the average scores for the two groups are the same.

Nick Johnson
Nick Johnson
Numerade Educator
01:53

Problem 68

Because of Facebook's rapid rise in popularity among college students, there is a great deal of interest in the relationship between Facebook use and academic performance. One study collected information on $n=1839$ undergraduate students to look at the relationships among frequency of Facebook use, participation in Facebook activities, time spent preparing for class, and overall
GPA. 26
Students reported preparing for class an average of 706 minutes per week with a standard deviation of 526 minutes. Students also reported spending an average of 106 minutes per day on Facebook with a standard deviation of 93 minutes; $8 \%$ of the students reported spending no time on Facebook.
(a) Construct a $95 \%$ confidence interval for the average number of minutes per week a student prepares for class.
(b) Construct a $95 \%$ confidence interval for the average number of minutes per week a student spends on Facebook. (Hint: Be sure to convert from minutes per day to minutes per week.)
(c) Explain why you might expect the population distributions of these two variables to be highly skewed to the right. Do you think this fact makes your confidence intervals invalid? Explain your answer.

Lucas Finney
Lucas Finney
Numerade Educator
00:53

Problem 69

Refer to the previous exercise. The authors state:
All students surveyed were U.S. residents admitted through the regular admissions process at a 4-year, public, primarily residential institution in the northeastern United States $(N=3866)$. Students were sent a link to a survey hosted on SurveyMonkey.com, a survey-hosting website, through their university-sponsored email accounts. For the students who did not participate immediately, two additional reminders were sent, 1 week apart. Participants were offered a chance to enter a drawing to win one of $90 \$ 10$ Amazon.com gift cards as incentive. A total of 1839 surveys were completed for an overall response rate of $48 \%$.
Discuss how these factors influence your interpretation of the results of this survey.

Trent Speier
Trent Speier
Numerade Educator
03:06

Problem 70

Refer to Exercise 7.68. Suppose that you wanted to compare the average minutes per week spent on Facebook with the average minutes per week spent preparing for class.
(a) Provide an estimate of this difference.
(b) Explain why it is incorrect to use the two-sample $t$ test to see if the means differ.

Beth Stone
Beth Stone
Numerade Educator
11:20

Problem 71

The "misery is not miserly" phenomenon refers to a person's spending judgment going haywire when the person is sad. In a study, 31 young adults were given $\$ 10$ and randomly assigned to either a sad or a neutral group. The participants in the sad group watched a video about the death of a boy's mentor (from The Champ), and those in the neutral group watched a video on the Great Barrier Reef.
After the video, each participant was offered the chance to trade $\$ 0.50$ increments of the $\$ 10$ for an insulated water bottle. ${ }^{27}$ Here are the data: SADNESS
(a) Examine each group's prices graphically. Is use of the $t$ procedures appropriate for these data? Carefully explain your answer.
(b) Make a table with the sample size, mean, and standard deviation for each of the two groups.
(c) State appropriate null and alternative hypotheses for comparing these two groups.
(d) Perform the significance test at the $\alpha=0.05$ level, making sure to report the test statistic, degrees of freedom, and $P$-value. What is your conclusion?
(e) Construct a $95 \%$ confidence interval for the mean difference in purchase price between the two groups.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:58

Problem 72

Traditional brand research argues that successful logos are ones that are highly relevant to the product they represent. However, a market research firm recently reported that nearly $20 \%$ of all table wine brands introduced in the last three years feature an animal on the label. Since animals have little to do with the product, why are marketers using this tactic?

Some researchers have proposed that consumers who are "primed" (in other words, they've thought about the image earlier in an unrelated context) process visual information more easily. 28 To demonstrate this, the researchers randomly assigned participants to either a primed or a nonprimed group. Each participant was asked to indicate his or her attitude toward a product on a seven-point scale (from 1 = dislike very much to 7 = like very much). A bottle of MagicCoat pet shampoo, with a picture of a collie on the label, was the product. Prior to giving this score, however, participants were asked to do a word find where four of the words were common to both groups (pet, grooming, bottle, label) and four were either related to the product image (dog, collie, puppy, woof) or conflicted with the image (cat, feline, kitten, meow). The following table contains the responses listed from smallest to largest. (.1. BPREF

Jameson Kuper
Jameson Kuper
Numerade Educator
04:12

Problem 73

QSRMagazine.com assessed 2053 drive-thru visits at quick-service restaurants. ${ }^{29}$ One benchmark assessed was customer service. Responses ranged from "Rude (1)" to "Very Friendly (5)." The following table breaks down the responses according to two of the chains studied. DRVTHRU
\begin{tabular}{lccccc}
\hline & \multicolumn{5}{c}{ Rating } \\
\cline { 2 - 6 } Chain & 1 & 2 & 3 & 4 & 5 \\
\hline Taco Bell & 5 & 3 & 54 & 109 & 136 \\
McDonald's & 2 & 22 & 73 & 165 & 100 \\
\hline
\end{tabular}
(a) Comment on the appropriateness of $t$ procedures for these data.
(b) Report the means and standard deviations of the ratings for each chain separately.
(c) Test whether the two chains, on average, have the same customer satisfaction. Use a two-sided alternative hypothesis and a significance level of $5 \%$.
(d) Construct a $95 \%$ confidence interval for the difference in average satisfaction.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
07:06

Problem 74

Researchers were interested in comparing the long-term psychological effects of being on a highcarbohydrate, low-fat (LF) diet versus a high-fat, low-carbohydrate (LC) diet. ${ }^{30}$ A total of 106 overweight and obese participants were randomly assigned to one of these two energy-restricted diets. At 52 weeks, 32 LC dieters and 33 LF dieters remained. Mood was assessed using a total mood disturbance score (TMDS), where a lower score is associated with a less negative mood. A summary of these results follows:
\begin{tabular}{lccc}
\hline Group & $n$ & $\mathrm{x}^{-}$ & $s$ \\
\hline LC & 32 & $47.3$ & $28.3$ \\
LF & 33 & $19.3$ & $25.8$ \\
\hline
\end{tabular}
(a) Is there a difference in the TMDS at Week 52 ? Test the null hypothesis that the dieters' average mood in the two groups is the same. Use a significance level of $0.05$.
(b) Critics of this study focus on the specific LC diet (that is, the science) and the dropout rate. Explain why the dropout rate is important to consider when drawing conclusions from this study.

Gus Steppen
Gus Steppen
Numerade Educator
03:45

Problem 75

Refer to Example 7.16 (page 456). That study also broke down the dietary composition of the main meal. The following table summarizes the total fats, protein, and carbohydrates in the main meal $(\mathrm{g}$ ) for the two groups:
(a) Is it appropriate to use the two-sample $t$ procedures that we studied in this section to analyze these data for group differences? Give reasons for your answer.
(b) Describe appropriate null and alternative hypotheses for comparing the two groups in terms of fats consumed.
(c) Carry out the significance test using $\alpha=0.05$. Report the test statistic with the degrees of freedom and the $P$-value. Write a short summary of your conclusion.
(d) Find a 95\% confidence interval for the difference between the two means. Compare the information given by the interval with the information given by the significance test.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
00:39

Problem 76

Refer to the previous exercise. Repeat parts (b) through (d) for protein and carbohydrates. Write a short summary of your findings.

Liuxi Sun
Liuxi Sun
Numerade Educator
02:31

Problem 77

Exposure to dust at work can lead to lung disease later in life. One study measured the workplace exposure of tunnel construction workers. ${ }^{31}$ Part of the study compared 115 drill and blast workers with 220 outdoor concrete workers. Total dust exposure was measured in milligram years per cubic meter $\left(\mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3\right)$. The mean exposure for the drill and blast workers was $18.0 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$ with a standard deviation of $7.8 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$. For the outdoor concrete workers, the corresponding values were $6.5 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$ and $3.4 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$.
(a) The sample included all workers for a tunnel construction company who received medical examinations as part of routine health checkups. Discuss the extent to which you think these results apply to other similar types of workers.
(b) Use a 95\% confidence interval to describe the difference in the exposures. Write a sentence that gives the interval and provides the meaning of $95 \%$ confidence.
(c) Test the null hypothesis that the exposures for these two types of workers are the same. Justify your choice of a one-sided or two-sided alternative. Report the test statistic, the degrees of freedom, and the $P$-value. Give a short summary of your conclusion.
(d) The authors of the article describing these results note that the distributions are somewhat skewed. Do you think that this fact makes your analysis invalid? Give reasons for your answer.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
02:17

Problem 78

Not all dust particles that are in the air around us cause problems for our lungs. Some particles are too large and stick to other areas of our body before they can get to our lungs. Others are so small that we can breathe them in and out and they will not deposit in our lungs. The researchers in the study described in the previous exercise also measured respirable dust. This is dust that deposits in our lungs when we breathe it. For the drill and blast workers, the mean exposure to respirable dust was $6.3 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$ with a standard deviation of $2.8 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$. The corresponding values for the outdoor concrete workers were $1.4 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$ and $0.7 \mathrm{mg} \cdot \mathrm{y} / \mathrm{m}^3$. Analyze these data using the questions in the previous exercise as a guide.

James Kiss
James Kiss
Numerade Educator
01:08

Problem 79

A study of food portion sizes reported that over a 17-year period, the average size of a soft drink consumed by Americans aged 2 years and older increased from $13.1$ ounces $(\mathrm{oz})$ to $19.9 \mathrm{oz}$. The authors state that the difference is statistically significant with $P<0.01 .^{32}$ Explain what additional information you would need to compute a confidence interval for the increase, and outline the procedure that you would use for the computations. Do you think that a confidence interval would provide useful additional information? Explain why or why not.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:52

Problem 80

The results in the previous exercise were based on two national surveys with a very large number of individuals. Here is a study that also looked at beverage consumption, but the sample sizes were much smaller. One part of this study compared 20 children who were 7 to 10 years old with 5 children who were 11 to $13.33$ The younger children consumed an average of $8.2$ oz of sweetened drinks per day while the older ones averaged $14.5 \mathrm{oz}$. The standard deviations were $10.7 \mathrm{oz}$ and $8.2$ oz, respectively.
(a) Do you think that it is reasonable to assume that these data are Normally distributed? Explain why or why not. (Hint: Think about the 68-95-99.7 rule.)
(b) Using the methods in this section, test the null hypothesis that the two groups of children consume equal amounts of sweetened drinks versus the two-sided alternative. Report all details of the significance-testing procedure with your conclusion.
(c) Give a $95 \%$ confidence interval for the difference in means.
(d) Do you think that the analyses performed in parts (b) and (c) are appropriate for these data? Explain why or why not.
(e) The children in this study were all participants in an intervention study at the Cornell Summer Day Camp at Cornell University. To what extent do you think that these results apply to other groups of children?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:08

Problem 81

Recall Exercise $7.58$ (page 455). You are concerned that day of the week may affect the number of hits. So to compare the two MySpace page designs, you choose two successive weeks in the middle of a month. You flip a coin to assign one Monday to the first design and the other Monday to the second. You repeat this for each of the seven days of the week. You now have 7 hit amounts for each design. It is incorrect to use the two-sample $t$ test to see if the mean hits differ for the two designs. Carefully explain why.

Carson Merrill
Carson Merrill
Numerade Educator
01:21

Problem 82

The purchasing department has suggested that all new computer monitors for your company should be flat screens. You want data to assure you that employees will like the new screens. The next 20 employees needing a new computer are the subjects for an experiment.
(a) Label the employees 01 to 20 . Randomly choose 10 to receive flat screens. The remaining 10 get standard monitors.
(b) After a month of use, employees express their satisfaction with their new monitors by responding to the statement "I like my new monitor" on a scale from 1 to 5 , where 1 represents "strongly disagree," 2 is "disagree," 3 is "neutral," 4 is "agree," and 5 stands for "strongly agree." The employees with the flat screens have average satisfaction $4.8$ with standard deviation $0.7$. The employees with the standard monitors have average $3.0$ with standard deviation $1.5$. Give a $95 \%$ confidence interval for the difference in the mean satisfaction scores for all employees.
(c) Would you reject the null hypothesis that the mean satisfaction for the two types of monitors is the same versus the two-sided alternative at significance level $0.05$ ? Use your confidence interval to answer this question. Explain why you do not need to calculate the test statistic.

Abdullah Alomair
Abdullah Alomair
Numerade Educator
02:27

Problem 83

Refer to the previous exercise. A coworker suggested that you give the flat screens to the next 10 employees who need new screens and the standard monitor to the following 10. Explain why your randomized design is better.

James Kiss
James Kiss
Numerade Educator
02:44

Problem 84

Corporate advertising tries to enhance the image of the corporation. A study compared two ads from two sources, the Wall Street Journal and the National Enquirer. Subjects were asked to pretend that their company was considering a major investment in Performax, the fictitious sportswear firm in the ads. Each subject was asked to respond to the question "How trustworthy was the source in the sportswear company ad for Performax?" on a 7-point scale. Higher values indicated more trustworthiness. ${ }^{34}$ Here is a summary of the results:
$$
\begin{array}{lccc}
\hline \text { Ad source } & n & \mathrm{x}^{-} & s \\
\hline \text { Wall Street Journal } & 66 & 4.77 & 1.50 \\
\text { National Enquirer } & 61 & 2.43 & 1.64 \\
\hline
\end{array}
$$
(a) Compare the two sources of ads using a $t$ test. Be sure to state your null and alternative hypotheses, the test statistic with degrees of freedom, the $P$-value, and your conclusion.
(b) Give a $95 \%$ confidence interval for the difference.
(c) Write a short paragraph summarizing the results of your analyses.

James Kiss
James Kiss
Numerade Educator
04:27

Problem 85

The study of 584 longleaf pine trees in the Wade Tract in Thomas County, Georgia, had several purposes. Are trees in one part of the tract more or less like trees in any other part of the tract or are there differences? In Example $6.1$ (page 352) we examined how the trees were distributed in the tract and found that the pattern was not random. In this exercise we will examine the sizes of the trees. In Exercise $7.31$ (page 443) we analyzed the sizes, measured as diameter at breast height (DBH), for a random sample of 40 trees. Here we divide the tract into northern and southern halves and take random samples of 30 trees from each half. Here are the diameters in centimeters (cm) of the sampled trees:
(a) Use a back-to-back stemplot and side-by-side boxplots to examine the data graphically. Describe the patterns in the data.
(b) Is it appropriate to use the methods of this section to compare the mean DBH of the trees in the north half of the tract with the mean DBH of the trees in the south half? Give reasons for your answer.
(c) What are appropriate null and alternative hypotheses for comparing the two samples of tree DBHs? Give reasons for your choices.
(d) Perform the significance test. Report the test statistic, the degrees of freedom, and the $P$-value. Summarize your conclusion.
(e) Find a 95\% confidence interval for the difference in mean DBHs. Explain how this interval provides additional information about this problem.

Carolyn Behr-Jerome
Carolyn Behr-Jerome
Numerade Educator
01:26

Problem 86

Refer to the previous exercise. The Wade Tract can also be divided into eastern and western halves.
Here are the DBHs of 30 randomly selected longleaf pine trees from each half:
EWPINES
Using the questions in the previous exercise, analyze these data.

Tony Wilson
Tony Wilson
Numerade Educator
01:04

Problem 87

A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimate and ignore sampling error. Suppose that an SRS of 70 stores this month shows mean sales of 53 units of a small appliance, with standard deviation 12 units. During the same month last year, an SRS of 55 stores gave mean sales of 50 units, with standard deviation 10 units. An increase from 50 to 53 is a rise of $6 \%$. The marketing manager is happy because sales are up $6 \%$.
(a) Use the two-sample $t$ procedure to give a $95 \%$ confidence interval for the difference in mean number of units sold at all retail stores.
(b) Explain in language that the manager can understand why he cannot be certain that sales rose by $6 \%$, and that in fact sales may even have dropped.

Tyler Moulton
Tyler Moulton
Numerade Educator
03:45

Problem 88

A friend has performed a significance test of the null hypothesis that two means are equal. His report states that the null hypothesis is rejected in favor of the alternative that the first mean is larger than the second. In a presentation on his work, he notes that the first sample mean was larger than the second mean and this is why he chose this particular one-sided alternative.
(a) Explain what is wrong with your friend's procedure and why.
(b) Suppose that he reported $t=1.70$ with a $P$-value of $0.06$. What is the correct $P$-value that he should report?

Jon Southam
Jon Southam
Numerade Educator
01:49

Problem 89

A study of iron deficiency among infants compared samples of infants following different feeding regimens. One group contained breast-fed infants, while the infants in another group were fed a standard baby formula without any iron supplements. Here are summary results on blood hemoglobin levels at 12 months of age: 35
(a) Is there significant evidence that the mean hemoglobin level is higher among breast-fed babies? State $H_0$ and $H_a$ and carry out a $t$ test. Give the $P$-value. What is your conclusion?
(b) Give a $95 \%$ confidence interval for the mean difference in hemoglobin level between the two populations of infants.
(c) State the assumptions that your procedures in parts (a) and (b) require in order to be valid.

Bryan Meares
Bryan Meares
Numerade Educator
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Problem 90

In Exercise $7.71$ (page 468), the purchase price of a water bottle was analyzed using the two-sample $t$ procedures that do not assume equal standard deviations. Compare the means using a significance test and find the $95 \%$ confidence interval for the difference using the pooled methods. How do the results compare with those you obtained in Exercise 7.71?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 91

In Exercise $7.72$ (page 469), attitudes toward a product were compared using the two-sample $t$ procedures that do not assume equal standard deviations. Compare the means using a significance test and find the $95 \%$ confidence interval for the difference using the pooled methods. How do the results compare with those you obtained in Exercise 7.72?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 92

In Exercise $7.75$ (page 469), the total amount of fats was analyzed using the two-sample $t$ procedures that do not assume equal standard deviations. Examine the standard deviations for the two groups and verify that it is appropriate to use the pooled procedures for these data. Compare the means using a significance test and find the $95 \%$ confidence interval for the difference using the pooled methods. How do the results compare with those you obtained in Exercise 7.75?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 93

Refer to the Wade Tract DBH data in Exercise $7.85$ (page 471), where we compared a sample of trees from the northern half of the tract with a sample from the southern half. Because the standard deviations for the two samples are quite close, it is reasonable to analyze these data using the pooled procedures. Perform the significance test and find the $95 \%$ confidence interval for the difference in means using these methods. Summarize your results and compare them with what you found in Exercise 7.85. NSPINES

Victor Salazar
Victor Salazar
Numerade Educator
01:54

Problem 94

Example 7.16 (page 456) gives summary statistics for weight loss in early eaters and late eaters. The two sample standard deviations are quite similar, so we may be willing to assume equal population standard deviations. Calculate the pooled $t$ test statistic and its degrees of freedom from the summary statistics. Use Table D to assess significance. How do your results compare with the unpooled analysis in the example?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:38

Problem 95

Use the Wade Tract data in Exercise $7.85$ to calculate the software approximation to the degrees of freedom using the formula on page 460 . Verify your calculation with software.

Aditya Sood
Aditya Sood
Numerade Educator
04:38

Problem 96

Use the Wade Tract data in Exercise $7.86$ to calculate the software approximation to the degrees of freedom using the formula on page 460 . Verify your calculation with software.

Aditya Sood
Aditya Sood
Numerade Educator
16:15

Problem 97

The data on occupational exposure to dust that we analyzed in Exercise $7.77$ (page 470 ) come from two groups of workers that are quite different in size. This complicates the issue regarding pooling because the sample that is larger will dominate the calculations.
(a) Calculate the software approximation to the degrees of freedom using the formula on page 460 . Then verify your calculations with software.
(b) Find the pooled estimate of the standard deviation. Write a short summary comparing it with the estimates of the standard deviations that come from each group.
(c) Find the standard error of the difference in sample means that you would use for the method that does not assume equal variances. Do the same for the pooled approach. Compare these two estimates with each other.
(d) Perform the significance test and find the $95 \%$ confidence interval using the pooled methods. How do these results compare with those you found in Exercise 7.77?
(e) Exercise $7.78$ has data for the same workers but for respirable dust. Here the standard deviations differ more than those in Exercise $7.77$ do. Answer parts (a) through (d) for these data. Write a summary of what you have found in this exercise.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
02:42

Problem 98

Refer to Example $7.17$ (page 457 ). This is a case where the sample sizes are quite small. With only 5 observations per group, we have very little information to make a judgment about whether the population standard deviations are equal. The potential gain from pooling is large when the sample sizes are small. Assume that we will perform a two-sided test using the $5 \%$ significance level. EATER
(a) Find the critical value for the unpooled $t$ test statistic that does not assume equal variances. Use the minimum of $n_1-1$ and $n_2-1$ for the degrees of freedom.
(b) Find the critical value for the pooled $t$ test statistic.
(c) How does comparing these critical values show an advantage of the pooled test?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:30

Problem 99

The $F$ statistic $\mathrm{F}=\mathrm{s} 12 / \mathrm{s} 22$ is calculated from samples of size $n_1=13$ and $n_2=22$
(a) What is the upper critical value for this $F$ when using the $0.05$ significance level?
(b) In a test of equality of standard deviations against the two-sided alternative, this statistic has the value $F=2.45$. Is this value significant at the $5 \%$ level? Is it significant at the $10 \%$ level?

Willis James
Willis James
Numerade Educator
01:43

Problem 100

If you repeat the calculation in Example $7.23$ for other values of $\mu_1-\mu_2$ that are larger than 5, would you expect the power to be higher or lower than $0.7965$ ? Why?

Chai Santi
Chai Santi
Numerade Educator
01:12

Problem 101

If the true population standard deviation were $7.1$ instead of the $7.4$ hypothesized in Example 7.23, would the power for this new experiment be greater or smaller than $0.7965$ ? Explain.

James Kiss
James Kiss
Numerade Educator
02:52

Problem 102

Here are some summary statistics from two independent samples from Normal distributions:
\begin{tabular}{ccc}
\hline Sample & $n$ & $s^2$ \\
\hline 1 & 11 & $3.5$ \\
2 & 16 & $9.1$ \\
\hline
\end{tabular}
You want to test the null hypothesis that the two population standard deviations are equal versus the twosided altemative at the $5 \%$ significance level.
(a) Calculate the test statistic.
(b) Find the appropriate value from Table E that you need to perform the significance test.
(c) What do you conclude?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:45

Problem 103

Compare the standard deviations of weight loss in Example $7.16$ (page 456). Give the test statistic, the degrees of freedom, and the $P$-value. Write a short summary of your analysis, including comments on the assumptions for the test.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
05:10

Problem 104

Compare the standard deviations of fat intake in Exercise $7.75$ (page 469).
(a) Give the test statistic, the degrees of freedom, and the $P$-value. Write a short summary of your analysis, including comments on the assumptions for the test.
(b) Assume that the sample standard deviation for the late-eaters group is the value $8.2$ given in Exercise 7.75. How large would the standard deviation in the early-eaters group need to be to reject the null hypothesis of equal standard deviations at the $5 \%$ level?

Amany Waheeb
Amany Waheeb
Numerade Educator
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Problem 105

The two-sample problem in Exercise $7.77$ (page 470) compares drill and blast workers with outdoor concrete workers with respect to the total dust that they are exposed to in the workplace. Here it may be useful to know whether or not the standard deviations differ in the two groups. Perform the $F$ test and summarize the results. Are you concerned about the assumptions here? Explain why or why not.

Victor Salazar
Victor Salazar
Numerade Educator
01:55

Problem 106


Exercise $7.78$ (page 470) is similar to Exercise 7.77, but the response variable here is exposure to dust particles that can enter and stay in the lungs. Compare the standard deviations with a significance test and summarize the results. Be sure to comment on the assumptions.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
01:32

Problem 107


The diameters of trees in the Wade Tract for random samples selected from the north and south halves of the tract are compared in Exercise $7.85$ (page 471). Is there a statistically significant difference between the standard deviations for these two parts of the tract? Perform the significance test and summarize the results. Does the Normal assumption appear reasonable for these data? NSPINES

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
03:21

Problem 108


Tree diameters for the east and west halves of the Wade Tract are compared in Exercise $7.86$ (page 472).
Using the questions in the previous exercise as a guide, analyze these data. EWPINES

Victoria Dollar
Victoria Dollar
Numerade Educator
03:48

Problem 109


In Example $7.17$ (page 457), we addressed a study with only 5 observations per group. EATER
(a) Is there a statistically significant difference between the standard deviations of these two groups?
Perform the test using a significance level of $0.05$ and state your conclusion.
(b) Using Table E, state the value that the ratio of variances would need to exceed for us to reject the null hypothesis (at the $5 \%$ level) that the standard deviations are equal. Also, report this value for sample sizes of $n=4,3$, and 2 . What does this suggest about the power of this test when sample sizes are small?

Beth Stone
Beth Stone
Numerade Educator
00:49

Problem 110


In Exercise $7.85$ (page 471) DBH data for longleaf pine trees in two parts of the Wade Tract are compared. Suppose that you are planning a similar study in which you will measure the diameters of longleaf pine trees. Based on Exercise 7.85, you are willing to assume that the standard deviation for both halves is 20 $\mathrm{cm}$. Suppose that a difference in mean DBH of $10 \mathrm{~cm}$ or more would be important to detect. You will use a $t$ statistic and a two-sided altemative for the comparison.
(a) Find the power if you randomly sample 20 trees from each area to be compared.
(b) Repeat the calculations for 60 trees in each sample.
(c) If you had to choose between the 20 and 60 trees per sample, which would you choose? Give reasons for your answer.

Anna Miller
Anna Miller
Numerade Educator
01:07

Problem 111


Refer to the previous exercise. Find the two standard deviations from Exercise 7.85. Do the same for the data in Exercise 7.86, which is a similar setting. These are somewhat smaller than the assumed value that you used in the previous exercise. Explain why it is generally a better idea to assume a standard deviation that is larger than you expect than one that is smaller. Repeat the power calculations for some other reasonable values of $\sigma$ and comment on the impact of the size of $\sigma$ for planning the new study.

Lucas Finney
Lucas Finney
Numerade Educator
02:44

Problem 112


Refer to Exercise $7.84$ (page 471), where we compared trustworthiness ratings for ads from two different publications. Suppose that you are planning a similar study using two different publications that are not expected to show the differences seen when comparing the Wall Street Journal with the National Enquirer. You would like to detect a difference of $1.5$ points using a two-sided significance test with a $5 \%$ level of significance. Based on Exercise 7.84, it is reasonable to use $1.6$ as the value of the common standard deviation for planning purposes.
(a) What is the power if you use sample sizes similar to those used in the previous study-for example, 65 for each publication?
(b) Repeat the calculations for 100 in each group.
(c) What sample size would you recommend for the new study?

James Kiss
James Kiss
Numerade Educator
02:11

Problem 113


The scores of four senior roommates on the Law School Admission Test (LSAT) are
$$
\begin{array}{llll}
156 & 133 & 147 & 122
\end{array}
$$
Find the mean, the standard deviation, and the standard error of the mean. Is it appropriate to calculate a confidence interval based on these data? Explain why or why not. Dlid LSAT

Heena Haldankar
Heena Haldankar
Numerade Educator
01:25

Problem 114


You use statistical software to perform a significance test of the null hypothesis that two means are equal. The software reports a $P$-value for the two-sided alternative. Your alternative is that the first mean is greater than the second mean.
(a) The software reports $t=2.08$ with a $P$-value of $0.068$. Would you reject $H_0$ at $\alpha$ Explain your answer.
(b) The software reports $t=-2.08$ with a $P$-value of $0.068$. Would you reject $H_0$ at $\alpha=0.05$ ?
Explain your answer.

Tyler Moulton
Tyler Moulton
Numerade Educator
03:50

Problem 115


As the degrees of freedom increase, the $t$ distributions get closer and closer to the $z(N(0,1))$ distribution. One way to see this is to look at how the value of $t^*$ for a $95 \%$ confidence interval changes with the degrees of freedom. Make a plot with degrees of freedom from 2 to 100 on the $x$ axis and $t^*$ on the $y$ axis. Draw a horizontal line on the plot corresponding to the value of $z^*=1.96$. Summarize the main features of the plot.

Amany Waheeb
Amany Waheeb
Numerade Educator
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Problem 116


Refer to the previous exercise. Make a similar plot for a $90 \%$ confidence interval. How do the main features of this plot compare with those of the plot in the previous exercise?

Victor Salazar
Victor Salazar
Numerade Educator
02:30

Problem 117


The margin of error for a confidence interval depends on the confidence level, the standard deviation, and the sample size. Fix the confidence level at $95 \%$ and the standard deviation at 1 to examine the effect of the sample size. Find the margin of error for sample sizes of 5 to 100 by $5 \mathrm{~s}-$ that is, let $n=5,10,15, \ldots, 100$. Plot the margins of error versus the sample size and summarize the relationship.

Lucas Finney
Lucas Finney
Numerade Educator
05:19

Problem 118


Refer to the previous exercise. Make a similar plot and summarize its features for a $99 \%$ confidence interval.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
03:37

Problem 119


The following situations all require inference about a mean or means. Identify each as (1) a single sample, (2) matched pairs, or (3) two independent samples. Explain your answers.
(a) Your customers are college students. You are interested in comparing the interest in a new product that you are developing between those students who live in the dorms and those who live elsewhere.
(b) Your customers are college students. You are interested in finding out which of two new product labels is more appealing.
(c) Your customers are college students. You are interested in assessing their interest in a new product.

Caleb Huber
Caleb Huber
Numerade Educator
03:37

Problem 120


The following situations all require inference about a mean or means. Identify each as (1) a single sample, (2) matched pairs, or (3) two independent samples. Explain your answers.
(a) You want to estimate the average age of your store's customers.
(b) You do an SRS survey of your customers every year. One of the questions on the survey asks about customer satisfaction on a seven-point scale with the response 1 indicating "very dissatisfied" and 7 indicating "very satisfied." You want to see if the mean customer satisfaction has improved from last year.
(c) You ask an SRS of customers their opinions on each of two new floor plans for your store.

Caleb Huber
Caleb Huber
Numerade Educator
10:35

Problem 121


The results of a major city's restaurant inspections are available through its online newspaper. 38 Critical food violations are those that put patrons at risk of getting sick and must immediately be corrected by the restaurant. An SRS of $n=200$ inspections from the more than 16,000 inspections since January 2009 were collected, resulting in $\mathrm{x}^{-}=0.83$ violations and $s=0.95$ violations.
(a) Test the hypothesis that the average number of critical violations is less than $1.5$ using a significance level of $0.05$. State the two hypotheses, the test statistic, and $P$-value.
(b) Construct a $95 \%$ confidence interval for the average number of critical violations and summarize your result.
(c) Which of the two summaries (significance test versus confidence interval) do you find more helpful in this case? Explain your answer.
(d) These data are integers ranging from 0 to 9 . The data are also skewed to the right, with $70 \%$ of the values either a 0 or a 1 . Given this information, do you think use of the $t$ procedures is appropriate? Explain your answer.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
05:25

Problem 122

Consider the following data set. The data were actually collected in pairs, and each row represents a
pair.
$$
\begin{array}{cc}
\hline \text { Group 1 } & \text { Group 2 } \\
\hline 48.86 & 48.88 \\
50.60 & 52.63 \\
51.02 & 52.55 \\
47.99 & 50.94 \\
54.20 & 53.02 \\
50.66 & 50.66 \\
45.91 & 47.78 \\
48.79 & 48.44 \\
47.76 & 48.92 \\
51.13 & 51.63 \\
\hline
\end{array}
$$
(a) Suppose that we ignore the fact that the data were collected in pairs and mistakenly treat this as a two-sample problem. Compute the sample mean and variance for each group. Then compute the two-sample $t$ statistic, degrees of freedom, and $P$-value for the two-sided alternative.
(b) Now analyze the data in the proper way. Compute the sample mean and variance of the differences. Then compute the $t$ statistic, degrees of freedom, and $P$-value.
(c) Describe the differences in the two test results.

Robin Corrigan
Robin Corrigan
Numerade Educator
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Problem 123

Refer to the previous exercise. Perhaps an easier way to see the major difference in the two analysis approaches for these data is by computing $95 \%$ confidence intervals for the mean difference.
(a) Compute the $95 \%$ confidence interval using the two-sample $t$ confidence interval.
(b) Compute the $95 \%$ confidence interval using the matched pairs $t$ confidence interval.
(c) Compare the estimates (that is, the centers of the intervals) and margins of error. What is the major difference between the two approaches for these data?

Victor Salazar
Victor Salazar
Numerade Educator
02:58

Problem 124

Recall the drive-thru study in Exercise $7.73$ (page 469). Another benchmark that was measured was the service time. A summary of the results (in seconds) for two of the chains is shown below.
$$
\begin{array}{lccc}
\hline \text { Chain } & n & x^{-} & s \\
\hline \text { Taco Bell } & 307 & 149.69 & 35.7 \\
\text { McDonald's } & 362 & 188.83 & 42.8 \\
\hline
\end{array}
$$
(a) Is there a difference in the average service time between these two chains? Test the null hypothesis that the chains' average service time is the same. Use a significance level of $0.05$.
(b) Construct a 95\% confidence interval for the difference in average service time.
(c) Lex plans to go to Taco Bell and Sam to McDonald's. Does the interval in part (b) contain the difference in their service times that they're likely to encounter? Explain your answer.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
06:30

Problem 125

A study utilized the random roommate assignment process of a small college to investigate the interracial mix of friends among students in college. ${ }^{39}$ As part of this study, the researchers looked at 238 white students who were randomly assigned a roommate in their first year and recorded the proportion of their friends (not including the first-year roommate) who were black. The following table summarizes the results, broken down by roommate race, for the middle of the first and third years of college.
(a) Proportions are not Normally distributed. Explain why it may still be appropriate to use the $t$ procedures for these data.
(b) For each year, state the null and alternative hypotheses for comparing these two groups.
(c) For each year, perform the significance test at the $\alpha=0.05$ level, making sure to report the test statistic, degrees of freedom, and $P$-value.
(d) Write a one-paragraph summary of your conclusions from these two tests.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:34

Problem 126

Refer to the previous exercise. For each year, construct a $95 \%$ confidence interval for the difference in means $\mu_1-\mu_2$ and describe how these intervals can be used to test the null hypotheses in part (b) of the previous exercise.

Jameson Kuper
Jameson Kuper
Numerade Educator
02:00

Problem 127

Individuals who consume large amounts of alcohol do not use the calories from this source as efficiently as calories from other sources. One study examined the effects of moderate alcohol consumption on body composition and the intake of other foods. Fourteen subjects participated in a crossover design where they either drank wine for the first 6 weeks and then abstained for the next 6 weeks or vice versa. 40 During the period when they drank wine, the subjects, on average, lost $0.4$ kilograms $(\mathrm{kg})$ of body weight; when they did not drink wine, they lost an average of $1.1 \mathrm{~kg}$. The standard deviation of the difference between the weight lost under these two conditions is $8.6 \mathrm{~kg}$ During the wine period, they consumed an average of 2589 calories; with no wine, the mean consumption was 2575 . The standard deviation of the difference was 210 .
(a) Compute the differences in means and the standard errors for comparing body weight and caloric intake under the two experimental conditions.
(b) A report of the study indicated that there were no significant differences in these two outcome measures. Verify this result for each measure, giving the test statistic, degrees of freedom, and the $P$ value.
(c) One concem with studies such as this, with a small number of subjects, is that there may not be sufficient power to detect differences that are potentially important. Address this question by computing $95 \%$ confidence intervals for the two measures and discuss the information provided by the intervals.
(d) Here are some other characteristics of the study. The study periods lasted for 6 weeks. All subjects were males between the ages of 21 and 50 years who weighed between 68 and $91 \mathrm{~kg}$. They were all from the same city. During the wine period, subjects were told to consume two 135 milliliter (ml) servings of red wine per day and no other alcohol. The entire 6-week supply was given to each subject at the beginning of the period. During the other period, subjects were instructed to refrain from any use of alcohol. All subjects reported that they complied with these instructions except for three subjects, who said that they drank no more than three to four 12-ounce bottles of beer during the no-alcohol period. Discuss how these factors could influence the interpretation of the results.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:51

Problem 128

The assessment of computerized brain-training programs is a rapidly growing area of research. Researchers are now focusing on who this training benefits most, what brain functions can be best improved, and which products are most effective. One study looked at 487 community-dwelling adults aged 65 and older, each randomly assigned to one of two training groups. In one group, the participants used a computerized program for 1 hour per day. In the other, DVD-based educational programs were shown with quizzes following each video. The training period lasted 8 weeks. The response was the improvement in a composite score obtained from an auditory memory/attention survey given before and after the 8 weeks. 41 The results are summarized in the following table.
$$
\begin{array}{lclc}
\hline \text { Group } & n & x^{-} & s \\
\hline \text { Computer program } & 242 & 3.9 & 8.28 \\
\text { DVD program } & 245 & 1.8 & 8.33 \\
\hline
\end{array}
$$
(a) Given that there are other studies showing a benefit of computerized brain training, state the null and alternative hypotheses.
(b) Report the test statistic, its degrees of freedom, and the $P$-value. What is your conclusion using significance level $\alpha=0.05$ ?
(c) Can you conclude that this computerized brain training always improves a person's auditory memory better than the DVD program? If not, explain why.

Lucas Finney
Lucas Finney
Numerade Educator
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Problem 129

A central question in urban ecology is why some animals adapt well to the presence of humans and others do not. The following results summarize part of a study of the northem mockingbird (Mimus polyglottos) that took place on a campus of a large university. ${ }^{42}$ For 4 consecutive days, the same human approached a nest and stood 1 meter away for 30 seconds, placing his or her hand on the rim of the nest. On the 5th day, a new person did the same thing. Each day, the distance of the human from the nest when the bird flushed was recorded. This was repeated for 24 nests. The human intruder varied his or her appearance (that is, wore different clothes) over the 4 days. We report results for only Days 1,4 , and 5 here. The response variable is flush distance measured in meters.
$$
\begin{array}{ccc}
\hline \text { Day } & \text { Mean } & s \\
\hline 1 & 6.1 & 4.9 \\
4 & 15.1 & 7.3 \\
5 & 4.9 & 5.3 \\
\hline
\end{array}
$$
(a) Explain why this should be treated as a matched design.
(b) Unfortunately, the research article does not provide the standard error of the difference, only the standard error of the mean flush distance for each day. However, we can use the general addition rule for variances (page 275) to approximate it. If we assume that the correlation between the flush distance at Day 1 and Day 4 for each nest is $\rho=0.40$, what is the standard deviation for the difference in distance?
(c) Using your result in part (b), test the hypothesis that there is no difference in the flush distance across these two days. Use a significance level of $0.05$.
(d) Repeat parts (b) and (c) but now compare Day 1 and Day 5, assuming a correlation between flush distances for each nest of $\rho=0.30$.
(c) Write a brief summary of your conclusions.

Victor Salazar
Victor Salazar
Numerade Educator
01:20

Problem 130

In one study, 39 diners were given a free glass of cabernet sauvignon wine to accompany a French meal. ${ }^{43}$ Although the wine was identical, half the bottle labels claimed the wine was from California and the other half claimed it was from North Dakota. The following table summarizes the grams of entrée and wine consumed during the meal.
$$
\begin{array}{ccccc}
\hline & \text { Wine label } & n & \text { Mean } & \text { St. dev. } \\
\hline \text { Entrée California } & 24 & 499.8 & 87.2 \\
& \text { North Dakota } & 15 & 439.0 & 89.2 \\
\text { Wine } & \text { California } & 24 & 100.8 & 23.3 \\
& \text { North Dakota } & 15 & 110.4 & 9.0 \\
\hline
\end{array}
$$
Did the patrons who thought that the wine was from Califormia consume more? Analyze the data and write a report summarizing your work. Be sure to include details regarding the statistical methods you used, your assumptions, and your conclusions.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:52

Problem 131

In the previous study, diners were seated alone or in groups of two, three, four, and, in one case, nine (for a total of $n=16$ tables). Also, each table, not each patron, was randomly assigned a particular wine label. Does this information alter how you might do the analysis in the previous problem?
Explain your answer.

Terrisa R
Terrisa R
Numerade Educator
02:02

Problem 132

The data used in Exercises $7.31$ (page 443), 7.85, and $7.86$ (pages 471 and 472 ) were obtained by taking simple random samples from the 584 longleaf pine trees that were measured in the Wade Tract. The entire data set is given in the WADE data set. Find the $95 \%$ confidence interval for the mean DBH using the entire data set, and compare this interval with the one that you calculated in Exercise 7.31. Write a report about these data. Include comments on the effect of the sample size on the margin of error, the distribution of the data, the appropriateness of the Normality-based methods for this problem, and the generalizability of the results to other similar stands of longleaf pine or other kinds of trees in this area of the United States and other areas. W. WADE

Lucas Finney
Lucas Finney
Numerade Educator
02:17

Problem 133

Suppose that your state contains 85 school corporations and each corporation reports its expenditures per pupil. Is it proper to apply the one-sample $t$ method to these data to give a $95 \%$ confidence interval for the average expenditure per pupil? Explain your answer.

James Kiss
James Kiss
Numerade Educator
03:22

Problem 134

A study was performed to determine the prevalence of the female athlete triad (low energy availability, menstrual dysfunction, and low bone mineral density) in high school students. ${ }^{44}$ A total of 80 high school athletes and 80 sedentary students were assessed. The following table summarizes several measured characteristics:
(a) For each of the characteristics, test the hypothesis that the means are the same in the two groups. Use a significance level of $0.05$ for each test.
(b) Write a short report summarizing your results.

Beth Stone
Beth Stone
Numerade Educator
03:28

Problem 135

A retailer entered into an exclusive agreement with a supplier who guaranteed to provide all products at competitive prices. The retailer eventually began to purchase supplies from other vendors who offered better prices. The original supplier filed a legal action claiming violation of the agreement. In defense, the retailer had an audit performed on a random sample of invoices. For each audited invoice, all purchases made from other suppliers were examined and the prices were compared with those offered by the original supplier. For each invoice, the percent of purchases for which the altemate supplier offered a lower price than the original supplier was recorded. ${ }^{45}$ Here are the data:
$$
681007910010010010010010089100100
$$
Report the average of the percents with a $95 \%$ margin of error. Do the sample invoices suggest that the original supplier's prices are not competitive on the average? $1.1$ COMPETE

Kevin Morgan
Kevin Morgan
Numerade Educator
02:49

Problem 136

In a study of the effectiveness of weight-loss programs, 47 subjects who were at least $20 \%$ overweight took part in a group support program for 10 weeks. Private weighings determined each subject's weight at the beginning of the program and 6 months after the program's end. The matched pairs $t$ test was used to assess the significance of the average weight loss. The paper reporting the study said, "The subjects lost a significant amount of weight over time, $t(46)=4.68, p>0.01$." It is common to report the results of statistical tests in this abbreviated style. 46
(a) Why was the matched pairs statistic appropriate?
(b) Explain to someone who knows no statistics but is interested in weight-loss programs what the practical conclusion is.
(c) The paper follows the tradition of reporting significance only at fixed levels such as $\alpha=0.01$. In fact, the results are more significant than " $p>0.01 "$ suggests. What can you say about the $P$-value of the $t$ test?

Lucas Finney
Lucas Finney
Numerade Educator
02:22

Problem 137

Some research suggests that women perform better than men in school, but men score higher on standardized tests. Table $1.3$ (page 29) presents data on a measure of school performance, grade point average (GPA), and a standardized test, IQ, for 78 seventh-grade students. Do these data lend further support to the previously found gender differences? Give graphical displays of the data and describe the distributions. Use significance tests and confidence intervals to examine this question, and prepare a short report summarizing your findings. D. GRADES

Anand Jangid
Anand Jangid
Numerade Educator
03:00

Problem 138

Refer to the previous exercise. Although self-concept in this study was measured on a scale with values in the data set ranging from 20 to 80 , many prefer to think of this kind of variable as having only two possible values: low self-concept or high self-concept. Find the median of the self-concept scores in Table 1.3, and define those students with scores at or below the median to be low-selfconcept students and those with scores above the median to be high-self-concept students. Do highself-concept students have GPAs that differ from those of low-self-concept students? What about IQ? Prepare a report addressing these questions. Be sure to include graphical and numerical summaries and confidence intervals, and state clearly the details of significance tests. Whe GADES

Marc Lauzon
Marc Lauzon
Numerade Educator
04:25

Problem 139

On the morning of March 5, 1996, a train with 14 tankers of propane derailed near the center of the small Wisconsin town of Weyauwega. Six of the tankers were ruptured and burning when the 1700 residents were ordered to evacuate the town. Researchers study disasters like this so that effective relief efforts can be designed for future disasters. About half the households with pets did not evacuate all their pets. A study conducted after the derailment focused on problems associated with retrieval of the pets after the evacuation and characteristics of the pet owners. One of the scales measured "commitment to adult animals," and the people who evacuated all or some of their pets were compared with those who did not evacuate any of their pets. Higher scores indicate that the pet owner is more likely to take actions that benefit the pet. ${ }^{47}$ Here are the data summaries:
$$
\begin{array}{lccc}
\hline \text { Group } & n & \mathrm{x}^{-} & \mathrm{s} \\
\hline \text { Evacuated all or some pets } & 116 & 7.95 & 3.62 \\
\text { Did not evacuate any pets } & 125 & 6.26 & 3.56 \\
\hline
\end{array}
$$
Analyze the data and prepare a short report describing the results.

Lottie Adams
Lottie Adams
Numerade Educator
02:07

Problem 140

Do various occupational groups differ in their diets? A British study of this question compared 98 drivers and 83 conductors of London double-decker buses. 48 The conductors' jobs require more physical activity. The article reporting the study gives the data as "Mean daily consumption ( $t$ se)." Here are some of the study results:
$$
\begin{array}{lcc}
\hline & \text { Drivers } & \text { Conductors } \\
\hline \text { Total calories } & 2821 \pm 44 & 2844 \pm 48 \\
\text { Alcohol (grams) } & 0.24 \pm 0.06 & 0.39 \pm 0.11 \\
\hline
\end{array}
$$
(a) What does "se" stand for? Give $\mathrm{x}^{-}$and $s$ for each of the four sets of measurements.
(b) Is there significant evidence at the $5 \%$ level that conductors consume more calories per day than do drivers? Use the two-sample $t$ method to give a $P$-value, and then assess significance.
(c) How significant is the observed difference in mean alcohol consumption? Use two-sample $t$ methods to obtain the $P$-value.
(d) Give a $95 \%$ confidence interval for the mean daily alcohol consumption of London doubledecker bus conductors.
(c) Give a $99 \%$ confidence interval for the difference in mean daily alcohol consumption between drivers and conductors.

Jessica Waggener
Jessica Waggener
Numerade Educator
01:34

Problem 141

Use of the pooled two-sample $t$ test is justified in part (b) of the previous exercise. Explain why. Find the $P$-value for the pooled $t$ statistic, and compare it with your result in the previous exercise.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:42

Problem 142

The report cited in Exercise $7.140$ says that the distributions of alcohol consumption among the individuals studied are "grossly skew."
(a) Do you think that this skewness prevents the use of the two-sample $t$ test for equality of means? Explain your answer.
(b) Do you think that the skewness of the distributions prevents the use of the $F$ test for equality of standard deviations? Explain your answer.

Nick Johnson
Nick Johnson
Numerade Educator
04:24

Problem 143

In the READ data set, the response variable Post3 is to be compared for three methods of teaching reading. The Basal method is the standard, or control, method, and the two new methods are DRTA and Strat. We can use the methods of this chapter to compare Basal with DRTA and Basal with Strat. Note that to make comparisons among three treatments it is more appropriate to use the procedures that we will learn in Chapter 12 . D. READ
(a) Is the mean reading score with the DRTA method higher than that for the Basal method? Perform an analysis to answer this question, and summarize your results.
(b) Answer part (a) for the Strat method in place of DRTA.

James Kiss
James Kiss
Numerade Educator
03:12

Problem 144

Example $7.13$ (page 449 ) tells us that the mean height of 10 -year-old girls is $N(56.4,2.7$ ) and for boys it is $N(55.7,3.8)$. The null hypothesis that the mean heights of 10 -year-old boys and girls are equal is clearly false. The difference in mean heights is $56.4-55.7=0.7$ inch. Small differences such as this can require large sample sizes to detect. To simplify our calculations, let's assume that the standard deviations are the same, say $\sigma=3.2$, and that we will measure the heights of an equal number of girls and boys. How many would we need to measure to have a $90 \%$ chance of detecting the (true) alternative hypothesis?

Nick Johnson
Nick Johnson
Numerade Educator