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Statistics Unlocking the Power of Data

Robin H. Lock, Patti Frazer Lock, Kari Lock Morgan

Chapter 3

Confidence Intervals - all with Video Answers

Educators

Section 1

Sampling Distributions

00:58

Problem 1

Average household income for all houses in the US, using data from the US Census.

Lucas Finney
Lucas Finney
Numerade Educator
00:55

Problem 2

Correlation between height and weight for players on the 2014 Brazil World Cup Team, using data from all 23 players on the roster.

Lucas Finney
Lucas Finney
Numerade Educator
01:13

Problem 3

Proportion of people who use an electric toothbrush, using data from a sample of 300 adults.

Nick Johnson
Nick Johnson
Numerade Educator
00:52

Problem 4

Proportion of registered voters in a county who voted in the last election, using data from the county voting records.

Lucas Finney
Lucas Finney
Numerade Educator
00:35

Problem 5

Average number of television sets per household in North Carolina, using data from a sample of 1000 households.

Lucas Finney
Lucas Finney
Numerade Educator
00:58

Problem 6

Give the correct notation for the quantity described and give its value.
Proportion of families in the US who were homeless in 2010 . The number of homeless families $^{5}$ in 2010 was about 170,000 while the total number of families is given in the 2010 Census as 78 million.

Lucas Finney
Lucas Finney
Numerade Educator
01:00

Problem 7

Give the correct notation for the quantity described and give its value.
Average enrollment in charter schools in Illinois. In $2014,$ there were 148 charter schools in the state of Illinois $^{6}$ and the total number of students attending the charter schools was 59.388

Lucas Finney
Lucas Finney
Numerade Educator
00:38

Problem 8

Give the correct notation for the quantity described and give its value.
Proportion of US adults who own a cell phone. In a survey of 1006 US adults in $2014,90 \%$ said they had a cell phone. $^{7}$

Lucas Finney
Lucas Finney
Numerade Educator
00:46

Problem 9

Correlation between age and heart rate for patients admitted to an Intensive Care Unit. Data from the 200 patients included in the file ICUAdmissions gives a correlation of 0.037 .

Lucas Finney
Lucas Finney
Numerade Educator
00:47

Problem 10

Mean number of cell phone calls made or received per day by cell phone users. In a survey of 1917 cell phone users, the mean was 13.10 phone calls a day.

Lucas Finney
Lucas Finney
Numerade Educator
01:03

Problem 11

Correlation between points and penalty minutes for all 24 players with at least 10 games played for the $2014-2015$ Ottawa Senators $^{9}$ NHL hockey team. The data are given in Table 3.4 and the full data are available in the file OttawaSenators.

Lucas Finney
Lucas Finney
Numerade Educator
01:06

Problem 12

Refer to the sampling distributions given in Figure $3.5 .$ In each case, estimate the value of the population parameter and estimate the standard error for the sample statistic.
Figure 3.5 (a) shows sample proportions from samples of size $n=40$ from a population.

Lucas Finney
Lucas Finney
Numerade Educator
01:04

Problem 13

Refer to the sampling distributions given in Figure $3.5 .$ In each case, estimate the value of the population parameter and estimate the standard error for the sample statistic.
Figure $3.5(\mathrm{~b})$ shows sample means from samples of size $n=30$ from a population.

Lucas Finney
Lucas Finney
Numerade Educator
00:52

Problem 14

Refer to the sampling distributions given in Figure $3.5 .$ In each case, estimate the value of the population parameter and estimate the standard error for the sample statistic.
Figure $3.5(\mathrm{c})$ shows sample means from samples of size $n=100$ from a population.

Lucas Finney
Lucas Finney
Numerade Educator
00:56

Problem 15

Refer to the sampling distributions given in Figure $3.5 .$ In each case, estimate the value of the population parameter and estimate the standard error for the sample statistic.
Figure $3.5($ d) shows sample proportions from samples of size $n=180$ from a population.

Lucas Finney
Lucas Finney
Numerade Educator
01:21

Problem 16

Refer to the sampling distributions given in Figure 3.5. Several possible values are given for a sample statistic. In each case, indicate whether each value is (i) reasonably likely to occur from a sample of this size, (ii) unusual but might occur occasionally, or (iii) extremely unlikely to ever occur.
Using the sampling distribution shown in Figure $3.5($ a $)$, how likely are these sample proportions:
(a) $\hat{p}=0.1$
(b) $\hat{p}=0.35$
(c) $\hat{p}=0.6$

Lucas Finney
Lucas Finney
Numerade Educator
00:50

Problem 17

Refer to the sampling distributions given in Figure 3.5. Several possible values are given for a sample statistic. In each case, indicate whether each value is (i) reasonably likely to occur from a sample of this size, (ii) unusual but might occur occasionally, or (iii) extremely unlikely to ever occur.
Using the sampling distribution shown in Figure $3.5(\mathrm{~b})$, how likely are these sample means:
(a) $\bar{x}=70$
(b) $\bar{x}=100$
(c) $\bar{x}=140$

Lucas Finney
Lucas Finney
Numerade Educator
00:57

Problem 18

Refer to the sampling distributions given in Figure 3.5. Several possible values are given for a sample statistic. In each case, indicate whether each value is (i) reasonably likely to occur from a sample of this size, (ii) unusual but might occur occasionally, or (iii) extremely unlikely to ever occur.
Using the sampling distribution shown in Figure $3.5(\mathrm{c})$, how likely are these sample means:
(a) $\bar{x}=250$
(b) $\bar{x}=305$
(c) $\bar{x}=315$

Lucas Finney
Lucas Finney
Numerade Educator
00:52

Problem 19

Refer to the sampling distributions given in Figure 3.5. Several possible values are given for a sample statistic. In each case, indicate whether each value is (i) reasonably likely to occur from a sample of this size, (ii) unusual but might occur occasionally, or (iii) extremely unlikely to ever occur.
Using the sampling distribution shown in Figure $3.5(\mathrm{~d}),$ how likely are these sample proportions:
(a) $\hat{p}=0.72$
(b) $\hat{p}=0.88$
(c) $\hat{p}=0.95$

Lucas Finney
Lucas Finney
Numerade Educator
01:20

Problem 20

Downloading Apps for Your Smartphone A random sample of $n=461$ smartphone users in the US in January 2015 found that 355 of them have downloaded an app. $^{10}$
(a) Give notation for the parameter of interest, and define the parameter in this context.
(b) Give notation for the quantity that gives the best estimate and give its value.
(c) What would we have to do to calculate the parameter exactly?

Lucas Finney
Lucas Finney
Numerade Educator
01:27

Problem 21

How Many Apps for Your Smartphone? Exercise 3.20 describes a study about smartphone users in the US downloading apps for their smartphone. Of the $n=355$ smartphone users who had downloaded an app, the average number of apps downloaded was 19.7
(a) Give notation for the parameter of interest, and define the parameter in this context.
(b) Give notation for the quantity that gives the best estimate and give its value.
(c) What would we have to do to calculate the parameter exactly?

Nick Johnson
Nick Johnson
Numerade Educator
01:23

Problem 22

Socially Conscious Consumers In March 2015, a Nielsen global online survey "found that consumers are increasingly willing to pay more for socially responsible products."11 Over 30,000 people in 60 countries were polled about their purchasing habits, and $66 \%$ of respondents said that they were willing to pay more for products and services from companies who are committed to positive social and environmental impact. We are interested in estimating the proportion of all consumers willing to pay more. Give notation for the quantity we are estimating, notation for the quantity we are using to make the estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Lucas Finney
Lucas Finney
Numerade Educator
01:53

Problem 23

Florida Lakes Florida has over 7700 lakes. $^{12}$ We wish to estimate the correlation between the pH levels of all Florida lakes and the mercury levels of fish in the lakes. We see in Data 2.4 on page 71 that the correlation between these two variables for a sample of $n=53$ of the lakes is -0.575 .
(a) Give notation for the quantity we are estimating, notation for the quantity we use to make the estimate, and the value of the best estimate.
(b) Why is an estimate necessary here? What would we have to do to calculate the exact value of the quantity we are estimating?

Lucas Finney
Lucas Finney
Numerade Educator
02:29

Problem 24

Topical Painkiller Ointment The use of topical painkiller ointment or gel rather than pills for pain relief was approved just within the last few years in the US for prescription use only. ${ }^{13}$ Insurance records show that the average copayment for a month's supply of topical painkiller ointment for regular users is \$30. A sample of $n=75$ regular users found a sample mean copayment of $\$ 27.90$.
(a) Identify each of 30 and 27.90 as a parameter or a statistic and give the appropriate notation for each.
(b) If we take 1000 samples of size $n=75$ from the population of all copayments for a month's supply of topical painkiller ointment for regular users and plot the sample means on a dotplot, describe the shape you would expect to see in the plot and where it would be centered.
(c) How many dots will be on the dotplot you described in part (b)? What will each dot represent?

Lucas Finney
Lucas Finney
Numerade Educator
03:21

Problem 25

Average Household Size The latest US Census lists the average household size for all households in the US as 2.61. (A household is all people occupying a housing unit as their primary place of residence.) Figure 3.6 shows possible distributions of means for 1000 samples of household sizes. The scale on the horizontal axis is the same in all four cases.
(a) Assume that two of the distributions show results from 1000 random samples, while two others show distributions from a sampling method that is biased. Which two dotplots appear to show samples produced using a biased sampling method? Explain your reasoning. Pick one of the distributions that you listed as biased and describe a sampling method that might produce this bias.
(b) For the two distributions that appear to show results from random samples, suppose that one comes from 1000 samples of size $n=100$ and one comes from 1000 samples of size $n=500$. Which distribution goes with which sample size? Explain.

Lucas Finney
Lucas Finney
Numerade Educator
05:15

Problem 26

5.26 Proportion of US Residents Less than 25 rears Old The US Census indicates that $35 \%$ of US residents are less than 25 years old. Figure 3.7 shows possible sampling distributions for the proportion of a sample less than 25 years old, for samples of size $n=20, n=100,$ and $n=500$
(a) Which distribution goes with which sample size?
(b) If we use a proportion $\hat{p},$ based on a sample of size $n=20,$ to estimate the population parameter $p=0.35,$ would it be very surprising to get an estimate that is off by more than 0.10 (that is, the sample proportion is less than 0.25 or greater than 0.45$)$ ? How about with a sample of size $n=100 ?$ How about with a sample of size $n=500 ?$
(c) Repeat part (b) if we ask about the sample proportion being off by just 0.05 or more.
(d) Using parts (b) and (c), comment on the effect that sample size has on the accuracy of an estimate.

Lucas Finney
Lucas Finney
Numerade Educator
02:23

Problem 27

Mix It Up for Better Learning In preparing for a test on a set of material, is it better to study one topic at a time or to study topics mixed together? In one study, $^{14}$ a sample of fourth graders were taught four equations. Half of the children learned by studying repeated examples of one equation at a time, while the other half studied mixed problem sets that included examples of all four types of calculations grouped together. A day later, all the students were given a test on the material. The students in the mixed practice group had an average grade of $77,$ while the students in the one-ata-time group had an average grade of $38 .$ What is the best estimate for the difference in the average grade between fourth-grade students who study mixed problems and those who study each equation independently? Give notation (as a difference with a minus sign) for the quantity we are trying to estimate, notation for the quantity that gives the best estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Lucas Finney
Lucas Finney
Numerade Educator
01:38

Problem 28

What Proportion of Adults and Teens Text Message? A study of $n=2252$ adults age 18 or older found that $72 \%$ of the cell phone users send and receive text messages. ${ }^{15}$ A study of $n=800$ teens age 12 to 17 found that $87 \%$ of the teen cell phone users send and receive text messages. What is the best estimate for the difference in the proportion of cell phone users who use text messages, between adults (defined as 18 and over) and teens? Give notation (as a difference with a minus sign) for the quantity we are trying to estimate, notation for the quantity that gives the best estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Nick Johnson
Nick Johnson
Numerade Educator
06:00

Problem 29

Hollywood Movies Data 2.7 on page 95 introduces the dataset HollywoodMovies, which contains information on more than 900 movies that came out of Hollywood between 2007 and $2013 .^{16}$ One of the variables is the budget (in millions of dollars) to make the movie. Figure 3.8 shows two boxplots. One represents the budget data for one random sample of size $n=30$. The other represents the values in a sampling distribution of 1000 means of budget data from samples of size 30 .
(a) Which is which? Explain.
(b) From the boxplot showing the data from one random sample, what does one value in the sample represent? How many values are included in the data to make the boxplot? Estimate the minimum and maximum values. Give a rough estimate of the mean of the values and use appropriate notation for your answer.
(c) From the boxplot showing the data from a sampling distribution, what does one value in the sampling distribution represent? How many values are included in the data to make the boxplot? Estimate the minimum and maximum values. Give a rough estimate of the value of the population parameter and use appropriate notation for your answer.

Lucas Finney
Lucas Finney
Numerade Educator
03:44

Problem 30

What Percent of the US Population Are Senior Citizens? People 65 years and older are the fastest growing segment of the US population, and constituted $13 \%$ of the population in $2010 .^{17}$ Figure 3.9 shows sample proportions from two sampling distributions of the proportion 65 years and older in the US: One shows samples of size 100 , and the other shows samples of size 1000 .
(a) What is the center of both distributions?
(b) What is the approximate minimum and maximum of each distribution?
(c) Give a rough estimate of the standard error in each case.
(d) Suppose you take one more sample in each case. Would a sample proportion of 0.17 (that is, $17 \%$ senior citizens in the sample) be surprising to see from a sample of size $100 ?$ Would it be surprising from a sample of size $1000 ?$

Lucas Finney
Lucas Finney
Numerade Educator
05:58

Problem 31

Graduate Programs in Statistics! One of the many wonderful things about studying statistics is that graduate programs in statistics often pay their graduate students, which means that many graduate students in statistics are able to attend graduate school tuition free with an assistantship or fellowship. In $2009,$ there were 82 US statistics or biostatistics doctoral programs for which enrollment data were available. ${ }^{18}$ The dataset StatisticsPhD lists all these schools together with the total enrollment of full-time graduate students in each program in $2009 .$
(a) Use StatKey or other technology to select a random sample of 10 of the 82 enrollment values. Indicate which values you've selected and compute the sample mean.
(b) Repeat part (a) by taking a second sample and calculating the mean.
(c) Find the mean enrollment for the entire population of these 82 graduate programs. Use correct notation for your answer. Comment on the accuracy of using the sample means found in parts (a) and (b) to estimate the population mean.
(d) Give a rough sketch of the sampling distribution if we calculate many sample means taking samples of size $n=10$ from this population of enrollment values. What shape will it have and where will it be centered?

Lucas Finney
Lucas Finney
Numerade Educator
03:50

Problem 32

Average Salary of NFL Players The dataset NFLContracts2015 contains the yearly salary (in millions of dollars) from the contracts of all players on a National Football League (NFL) roster at the start of the 2015 season. ${ }^{19}$
(a) Use StatKey or other technology to select a random sample of 5 NFL contract YearlySalary values. Indicate which players you've selected and compute the sample mean.
(b) Repeat part (a) by taking a second sample of 5 values, again indicating which players you selected and computing the sample mean.
(c) Find the mean for the entire population of players. Include notation for this mean. Comment on the accuracy of using the sample means found in parts (a) and (b) to estimate the population mean.

Lucas Finney
Lucas Finney
Numerade Educator
03:19

Problem 33

A Sampling Distribution for Statistics Graduate Programs Exercise 3.31 introduced the dataset StatisticsPhD, which gives enrollment for all 82 graduate statistics programs in the US in $2009 .$ Use StatKey or other technology to generate a sampling distribution of sample means using a sample size of $n=10$ from the values in this dataset. What shape does the distribution have? Approximately where is it centered? What is the standard error (in other words, what is the standard deviation of the sample means)?

Lucas Finney
Lucas Finney
Numerade Educator
04:31

Problem 34

3.34 A Sampling Distribution for Average Salary of NFL Players Use StatKey or other technology to generate a sampling distribution of sample means using a sample of size $n=5$ from the YearlySalary values in the dataset NFLContracts2015, which gives the total and yearly money values from the contracts of all NFL players in 2015 .
(a) What are the smallest and largest YearlySalary values in the population?
(b) What are the smallest and largest sample means in the sampling distribution?
(c) What is the standard error (that is, the standard deviation of the sample means) for the sampling distribution for samples of size $n=5 ?$
(d) Generate a new sampling distribution with samples of size $n=50 .$ What is the standard error for this sampling distribution?

Lucas Finney
Lucas Finney
Numerade Educator
04:49

Problem 35

What Is an Average Budget for a Hollywood Movie? Data 2.7 on page 95 introduces the dataset HollywoodMovies, which contains information on more than 900 movies that came out of Hollywood between 2007 and $2013 .$ We will consider this the population of all movies produced in Hollywood during this time period.
(a) Find the mean and standard deviation for the budgets (in millions of dollars) of all Hollywood movies between 2007 and $2013 .$ Use the correct notation with your answer.
(b) Use StatKey or other technology to generate a sampling distribution for the sample mean of budgets of Hollywood movies during this period using a sample size of $n=20$. Give the shape and center of the sampling distribution and give the standard error.

Lucas Finney
Lucas Finney
Numerade Educator
03:01

Problem 36

College Graduates In Example 3.1 on page 197, we see that $27.5 \%$ of US adults are college graduates.
(a) Use StatKey or other technology to generate a sampling distribution for the sample proportion of college graduates using a sample size of $n=50 .$ Generate at least 1000 sample proportions. Give the shape and center of the sampling distribution and give the standard error.
(b) Repeat part (a) using a sample size of $n=500$.

Lucas Finney
Lucas Finney
Numerade Educator
01:39

Problem 37

Gender in the Rock and Roll Hall of Fame From its founding through $2015,$ the Rock and Roll Hall of Fame has inducted 303 groups or individuals. Forty-seven of the inductees have been female or have included female members. $^{20}$ The full dataset is available in RockandRoll.
(a) What proportion of inductees have been female or have included female members? Use the correct notation with your answer.
(b) If we took many samples of size 50 from the population of all inductees and recorded the proportion female or with female members for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

Lucas Finney
Lucas Finney
Numerade Educator
02:24

Problem 38

Performers in the Rock and Roll Hall of Fame From its founding through $2015,$ the Rock and Roll Hall of Fame has inducted 303 groups or individuals, and 206 of the inductees have been performers while the rest have been related to the world of music in some way other than as a performer. The full dataset is available in RockandRoll.
(a) What proportion of inductees have been performers? Use the correct notation with your answer.
(b) If we took many samples of size 50 from the population of all inductees and recorded the proportion who were performers for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

Lucas Finney
Lucas Finney
Numerade Educator
04:32

Problem 39

A Sampling Distribution for Gender in the Rock and Roll Hall of Fame Exercise 3.37 tells us that 47 of the 303 inductees to the Rock and Roll Hall of Fame have been female or have included female members. The data are given in RockandRoll. Using all inductees as your population:
(a) Use StatKey or other technology to take many random samples of size $n=10$ and compute the sample proportion that are female or with female members. What is the standard error for these sample proportions? What is the value of the sample proportion farthest from the population proportion of $p=0.155 ?$ How far away is it?
(b) Repeat part (a) using samples of size $n=20$.
(c) Repeat part (a) using samples of size $n=50$.
(d) Use your answers to parts (a), (b), and (c) to comment on the effect of increasing the sample size on the accuracy of using a sample proportion to estimate the population proportion.

Lucas Finney
Lucas Finney
Numerade Educator
02:13

Problem 40

A Sampling Distribution for Performers in the Rock and Roll Hall of Fame Exercise 3.38 tells us that 206 of the 303 inductees to the Rock and Roll Hall of Fame have been performers. The data are given in RockandRoll. Using all inductees as your population:
(a) Use StatKey or other technology to take many random samples of size $n=10$ and compute the sample proportion that are performers. What is the standard error of the sample proportions? What is the value of the sample proportion farthest from the population proportion of $p=0.68 ?$ How far away is it?
(b) Repeat part (a) using samples of size $n=20$.
(c) Repeat part (a) using samples of size $n=50$.
(d) Use your answers to parts (a), (b), and (c) to comment on the effect of increasing the sample size on the accuracy of using a sample proportion to estimate the population proportion.

Nick Johnson
Nick Johnson
Numerade Educator