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Elementary Statistics a Step by Step Approach

Allan G. Bluman

Chapter 7

Confidence Intervals and Sample Size - all with Video Answers

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Section 1

Confidence Intervals for the Mean When ? Is Known

01:42

Problem 1

What is the difference between a point estimate and an interval estimate of a parameter? Which is better? Why?

Monique Whittaker
Monique Whittaker
Numerade Educator
02:21

Problem 2

What information is necessary to calculate a confidence interval?

Monique Whittaker
Monique Whittaker
Numerade Educator
02:25

Problem 3

What is the margin of error?

Monique Whittaker
Monique Whittaker
Numerade Educator
01:32

Problem 4

What is meant by the $95 \%$ confidence interval of the mean?

Monique Whittaker
Monique Whittaker
Numerade Educator
02:40

Problem 5

What are three properties of a good estimator?

Monique Whittaker
Monique Whittaker
Numerade Educator
00:34

Problem 6

What statistic best estimates $\mu ?$

Monique Whittaker
Monique Whittaker
Numerade Educator
01:40

Problem 7

Find each.
a. $z_{a / 2}$ for the $99 \%$ confidence interval
b. $z_{\alpha / 2}$ for the $98 \%$ confidence interval
c. $z_{a / 2}$ for the $95 \%$ confidence interval
d. $z_{\alpha / 2}$ for the $90 \%$ confidence interval
e. $z_{\alpha / 2}$ for the $94 \%$ confidence interval

Nick Johnson
Nick Johnson
Numerade Educator
01:28

Problem 8

What is necessary to determine the sample size?

Monique Whittaker
Monique Whittaker
Numerade Educator
01:17

Problem 9

Since 1975 the average fuel efficiency of U.S. cars and light trucks (SUVs) has increased from 13.5 to $25.8 \mathrm{mpg}$, an increase of over $90 \% !$ A random sample of 40 cars from a large community got a mean mileage of 28.1 mpg per vehicle. The population standard deviation is 4.7 mpg. Estimate the true mean gas mileage with $95 \%$ confidence.

Jameson Kuper
Jameson Kuper
Numerade Educator
02:03

Problem 10

A random sample of 50 cars in the drive-thru of a popular fast food restaurant revealed an average bill of $\$ 18.21$ per car. The population standard deviation is $\$ 5.92 .$ Estimate the mean bill for all cars from the drive-thru with $98 \%$ confidence.

Prashant Bana
Prashant Bana
Numerade Educator
01:55

Problem 11

For a random sample of 60 overweight men, the mean of the number of pounds that they were overweight was $30 .$ The standard deviation of the population is 4.2 pounds.
a. Find the best point estimate of the average number of excess pounds that they weighed.
b. Find the $95 \%$ confidence interval of the mean of these pounds.
c. Find the $99 \%$ confidence interval of these pounds.
d. Which interval is larger? Why?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
11:14

Problem 12

A sociologist found that in a random sample of 50 retired men, the average number of jobs they had during their lifetimes was $7.2 .$ The population standard deviation is 2.1
a. Find the best point estimate of the population mean.
b. Find the $95 \%$ confidence interval of the mean number of jobs.
c. Find the $99 \%$ confidence interval of the mean number of jobs.
d. Which is smaller? Explain why.

SB
Shannon Baxter
Numerade Educator
01:29

Problem 13

The numbers of faculty at 32 randomly selected state-controlled colleges and universities with enrollment under 12.000 students are shown below. Use these data to estimate the mean number of faculty at all state-controlled colleges and universities with enrollment under 12,000 with $92 \%$ confidence. Assume $\sigma=165.1$.
$\begin{array}{lllllllll}211 & 384 & 396 & 211 & 224 & 337 & 395 & 121 & 356 \\ 621 & 367 & 408 & 515 & 280 & 289 & 180 & 431 & 176 \\ 318 & 836 & 203 & 374 & 224 & 121 & 412 & 134 & 539 \\ 471 & 638 & 425 & 159 & 324 & & & & \end{array}$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:14

Problem 14

First-semester GPAs for a random selection of freshmen at a large university are shown. Estimate the true mean GPA of the freshman class with $99 \%$ confidence. Assume $\sigma=0.62$
$$
\begin{array}{llllll}
1.9 & 3.2 & 2.0 & 2.9 & 2.7 & 3.3 \\
2.8 & 3.0 & 3.8 & 2.7 & 2.0 & 1.9 \\
2.5 & 2.7 & 2.8 & 3.2 & 3.0 & 3.8 \\
3.1 & 2.7 & 3.5 & 3.8 & 3.9 & 2.7 \\
2.0 & 2.8 & 1.9 & 4.0 & 2.2 & 2.8 \\
2.1 & 2.4 & 3.0 & 3.4 & 2.9 & 2.1
\end{array}
$$

Monique Whittaker
Monique Whittaker
Numerade Educator
01:29

Problem 15

The number of grams of carbohydrates in various commercially prepared 7 -inch subs is recorded below. The population standard deviation is $6.46 .$ Estimate the mean number of carbs in all similarly sized subs with $95 \%$ confidence.
$$
\begin{array}{lllllllll}
63 & 67 & 61 & 64 & 51 & 42 & 56 & 70 & 61 \\
55 & 60 & 55 & 57 & 60 & 60 & 66 & 55 & 58 \\
70 & 65 & 49 & 51 & 61 & 54 & 50 & 55 & 56 \\
53 & 65 & 68 & 63 & 48 & 54 & 56 & 57 &
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:24

Problem 16

A random sample of the number of farms (in thousands) in various states follows. Estimate the mean number of farms per state with $90 \%$ confidence. Assume $\sigma=31.0$.
$$
\begin{aligned}
&\begin{array}{rrrrrrrrrr}
47 & 95 & 54 & 33 & 64 & 4 & 8 & 57 & 9 & 80 \\
8 & 90 & 3 & 49 & 4 & 44 & 79 & 80 & 48 & 16 \\
68 & 7 & 15 & 21 & 52 & 6 & 78 & 109 & 40 & 50
\end{array}\\
&29
\end{aligned}
$$

Monique Whittaker
Monique Whittaker
Numerade Educator
02:24

Problem 16

A random sample of the number of farms (in thousands) in various states follows. Estimate the mean number of farms per state with $90 \%$ confidence. Assume $\sigma=31.0$.
$$
\begin{aligned}
&\begin{array}{rrrrrrrrrr}
47 & 95 & 54 & 33 & 64 & 4 & 8 & 57 & 9 & 80 \\
8 & 90 & 3 & 49 & 4 & 44 & 79 & 80 & 48 & 16 \\
68 & 7 & 15 & 21 & 52 & 6 & 78 & 109 & 40 & 50
\end{array}\\
&29
\end{aligned}
$$

Monique Whittaker
Monique Whittaker
Numerade Educator
01:36

Problem 17

A random sample of 36 drivers used on average 749 gallons of gasoline per year. If the standard deviation of the population is 32 gallons, find the $95 \%$ confidence interval of the mean for all drivers. If a driver said that he used 803 gallons per year, would you believe that?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
View

Problem 18

A random sample of 50 four-year-olds attending day care centers provided a yearly tuition average of $\$ 3987$ and the population standard deviation of $\$ 630$. Find the $90 \%$ confidence interval of the true mean. If a day care center were starting up and wanted to keep tuition low, what would be a reasonable amount to charge?

Clarissa Barr
Clarissa Barr
Numerade Educator
01:51

Problem 19

Noise levels at various area urban hospitals were measured in decibels. The mean of the noise levels in 84 randomly selected corridors was 61.2 decibels, and the standard deviation of the population was 7.9 . Find the $95 \%$ confidence interval of the true mean.

Prashant Bana
Prashant Bana
Numerade Educator
02:52

Problem 20

The growing seasons for a random sample of 35 U.S. cities were recorded, yielding a sample mean of 190.7 days and the population standard deviation of 54.2 days. Estimate for all U.S. cities the true mean of the growing season with $95 \%$ confidence.

Sanchit Jain
Sanchit Jain
Numerade Educator
02:36

Problem 21

How large a sample is needed to estimate the population mean for the amount of money a person spends on Christmas presents within $\$ 2$ and be $95 \%$ confident? The standard deviation of the population is $\$ 7.50$.

Norman Atentar
Norman Atentar
Numerade Educator
02:44

Problem 22

In the hospital study cited in Exercise $19,$ the mean noise level in 171 randomly selected ward areas was 58.0 decibels, and the population standard deviation was 4.8 . Find the $90 \%$ confidence interval of the true mean.

Sanchit Jain
Sanchit Jain
Numerade Educator
02:31

Problem 23

A researcher wishes to estimate the average number of minutes per day a person spends on the Internet. How large a sample must she select if she wishes to be $90 \%$ confident that the population mean is within 10 minutes of the sample mean? Assume the population standard deviation is 42 minutes.

Sanchit Jain
Sanchit Jain
Numerade Educator
02:07

Problem 24

Cost of Pizzas A pizza shop owner wishes to find the $95 \%$ confidence interval of the true mean cost of a large cheese pizza. How large should the sample be if she wishes to be accurate to within $\$ 0.15 ?$ A previous study showed that the standard deviation of the price was $\$ 0.26$.

Monique Whittaker
Monique Whittaker
Numerade Educator
03:03

Problem 25

If the variance of the water temperature in a lake is $28^{\circ},$ how many days should the researcher select to measure the temperature to estimate the true mean within $3^{\circ}$ with $99 \%$ confidence?

Sanchit Jain
Sanchit Jain
Numerade Educator
01:32

Problem 26

It is desired to estimate the mean GPA of each undergraduate class at a large university. How large a sample is necessary to estimate the GPA within 0.25 at the $99 \%$ confidence level? The population standard deviation is $1.2 .$

Sanchit Jain
Sanchit Jain
Numerade Educator