Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 7

Confidence Interval Estimation: One Population - all with Video Answers

Educators

Chapter Questions

Problem 1

There is concern about the speed of automobiles traveling over a particular stretch of highway. For a random sample of 28 automobiles, radar indicated the following speeds, in miles per hour:

$\begin{array}{lllllll}59 & 63 & 68 & 57 & 56 & 71 & 59\end{array}$
$\begin{array}{lllllll}69 & 53 & 58 & 60 & 66 & 51 & 59\end{array}$
$\begin{array}{lllllll}54 & 64 & 58 & 57 & 66 & 61 & 65\end{array}$
$\begin{array}{lllllll}70 & 63 & 65 & 57 & 56 & 61 & 59\end{array}$

a. Check for evidence of nonnormality.
b. Find a point estimate of the population mean that is unbiased and efficient.
c. Use an unbiased estimation procedure to find a point estimate of the variance of the sample mean.

Sheryl Ezze

Sheryl Ezze

Numerade Educator

Problem 1

Calculate the margin of error to estimate the population mean for each of the following.
a. $99 \%$ confidence level;
$$
x_1=25 ; x_2=30 ; x_3=33 ; x_4=21
$$
b. $90 \%$ confidence level;
$$
x_1=15 ; x_2=17 ; x_3=13 ; x_4=11 ; x_5=14
$$

Sheryl Ezze

Numerade Educator

Problem 2

A random sample of eight homes in a particular suburb had the following selling prices (in thousands of dollars):
$\begin{array}{llllllll}192 & 183 & 312 & 227 & 309 & 396 & 402 & 390\end{array}$
a. Check for evidence of nonnormality.
b. Find a point estimate of the population mean that is unbiased and efficient.
c. Use an unbiased estimation procedure to find a point estimate of the variance of the sample mean.
d. Use an unbiased estimator to estimate the proportion of homes in this suburb selling for less than $$\$ 250,000$$.

Sheryl Ezze

Numerade Educator

Problem 3

A random sample of 10 economists produced the following forecasts for percentage growth in real gross domestic product in the next year:
$\begin{array}{llllllllll}2.2 & 2.8 & 3.0 & 2.5 & 2.4 & 2.6 & 2.5 & 2.4 & 2.7 & 2.6\end{array}$
Use unbiased estimation procedures to find point estimates for the following:
a. The population mean
b. The population variance
c. The variance of the sample mean
d. The population proportion of economists predicting growth of at least $2.5 \%$ in real gross domestic product

Sheryl Ezze

Numerade Educator

Problem 4

A random sample of 12 employees in a large manufacturing plant found the following figures for number of hours of overtime worked in the last month:
$\begin{array}{llllllllllll}22 & 16 & 28 & 12 & 18 & 36 & 23 & 11 & 41 & 29 & 26 & 31\end{array}$

Sheryl Ezze

Numerade Educator

Problem 5

Use unbiased estimation procedures to find point estimates for the following:
a. The population mean
b. The population variance
c. The variance of the sample mean
d. The population proportion of employees working more than 30 hours of overtime in this plant in the last month

Sheryl Ezze

Numerade Educator

Problem 6

The demand for bottled water increases during the hurricane season in Florida. The operations manager at a plant that bottles drinking water wants to be sure that the filling process for 1-gallon bottles (1 gallon is approximately 3.785 liters) is operating properly. Currently, the company is testing the volumes of one-gallon bottles. Suppose that a random sample of 75 bottles is tested, and the measurements are recorded in the data file Water.
a. Is there evidence that the data are not normally distributed?
b. Find a minimum variance unbiased point estimate of the population mean.
c. Find a minimum variance unbiased point estimate of the population variance.

Nick Johnson

Numerade Educator

Problem 7

Suppose that $x_1$ and $x_2$ are random samples of observations from a population with mean $\mu$ and variance $s^2$. Consider the following three point estimators, $X, Y$, $Z$, of $\mu$ :
$$
X=\frac{1}{2^2}+\frac{1}{2^2} \quad Y=\frac{1}{4} x_1+\frac{3}{4} x_2 \quad Z=\frac{1}{3} x_1+\frac{2}{3} x_2
$$
a. Show that all three estimators are unbiased.
b. Which of the estimators is the most efficient?
c. Find the relative efficiency of $X$ with respect to each of the other two estimators.

Victor Salazar

Numerade Educator

Problem 8

Find the reliability factor, $z_{\alpha / 2}$, to estimate the mean, $\mu$, of a normally distributed population with known population variance for the following.
a. $93 \%$ confidence level
b. $96 \%$ confidence level
c. $80 \%$ confidence level

Sheryl Ezze

Numerade Educator

Problem 9

Find the reliability factor, $z_{\alpha / 2}$, to estimate the mean, $\mu$, of a normally distributed population with known population variance for the following.
a. $\alpha=0.08$
b. $\alpha / 2=0.02$

Sheryl Ezze

Numerade Educator

Problem 10

Assume a normal distribution with known population variance. Calculate the margin of error to estimate the population mean, $\mu$, for the following.
a. $98 \%$ confidence level; $n=64 ; \sigma^2=144$
b. $99 \%$ confidence level; $n=120 ; \sigma=100$

Sheryl Ezze

Numerade Educator

Problem 11

Assume a normal distribution with known population variance. Calculate the width to estimate the population mean, $\mu$, for the following.
a. $90 \%$ confidence level; $n=100 ; \sigma^2=169$
b. $95 \%$ confidence level; $n=120 ; \sigma=25$

Sheryl Ezze

Numerade Educator

Problem 12

Assume a normal distribution with known population variance. Calculate the LCL and UCL for each of the following.
a. $\bar{x}=50 ; n=64 ; \sigma=40 ; \alpha=0.05$
b. $\bar{x}=85 ; n=225 ; \sigma^2=400 ; \alpha=0.01$
c. $\bar{x}=510 ; n=485 ; \sigma=50 ; \alpha=0.10$

Sheryl Ezze

Numerade Educator

Problem 13

A personnel manager has found that historically the scores on aptitude tests given to applicants for entry level positions follow a normal distribution with a standard deviation of 32.4 points. A random sample of nine test scores from the current group of applicants had a mean score of 187.9 points.

a. Find an $80 \%$ confidence interval for the population mean score of the current group of applicants.
b. Based on these sample results, a statistician found for the population mean a confidence interval extending from 165.8 to 210.0 points. Find the confidence level of this interval.

Sheryl Ezze

Numerade Educator

Problem 14

It is known that the standard deviation in the volumes of 20-ounce (591-milliliter) bottles of natural spring water bottled by a particular company is 5 millliliters. One hundred bottles are randomly sampled and measured.
a. Calculate the standard error of the mean.
b. Find the margin of error of a $90 \%$ confidence interval estimate for the population mean volume.
c. Calculate the width for a $98 \%$ confidence interval for the population mean volume.

Sheryl Ezze

Numerade Educator

Problem 15

A college admissions officer for an MBA program has determined that historically applicants have undergraduate grade point averages that are normally distributed with standard deviation 0.45 . From a random sample of 25 applications from the current year, the sample mean grade point average is 2.90 .
a. Find a $95 \%$ confidence interval for the population mean.
b. Based on these sample results, a statistician computes for the population mean a confidence interval extending from 2.81 to 2.99 . Find the confidence level associated with this interval.

Sheryl Ezze

Numerade Educator

Problem 16

A process produces bags of refined sugar. The weights of the contents of these bags are normally distributed with standard deviation 1.2 ounces. The contents of a random sample of 25 bags had a mean weight of 19.8 ounces. Find the upper and lower confidence limits of a $99 \%$ confidence interval for the true mean weight for all bags of sugar produced by the process.

Sheryl Ezze

Numerade Educator

Problem 17

Find the standard error to estimate the population mean for each of the following.
a. $n=17 ; 95 \%$ confidence level; $s=16$
b. $n=25 ; 90 \%$ confidence level; $s^2=43$

Sheryl Ezze

Numerade Educator

Problem 19

Twenty people in one large metropolitan area were asked to record the time (in minutes) that it takes them to drive to work. These times were as follows:
$\begin{array}{llllllllll}30 & 42 & 35 & 40 & 45 & 22 & 32 & 15 & 41 & 45 \\ 28 & 32 & 45 & 27 & 47 & 50 & 30 & 25 & 46 & 25\end{array}$
a. Calculate the standard error.
b. Find $t_{v, \alpha / 2}$ for a $95 \%$ confidence interval for the true population mean.
c. Calculate the width for a $95 \%$ confidence interval for the population mean time spent driving to work.

Sheryl Ezze

Numerade Educator

Problem 20

Find the LCL and UCL for each of the following.
a. $\alpha=0.05 ; n=25 ; \bar{x}=560 ; s=45$
b. $\alpha / 2=0.05 ; n=9 ; x=160 ; s^2=36$
c. $1-\alpha=0.98 ; n=22 ; \bar{x}=58 ; s=15$

Sheryl Ezze

Numerade Educator

Problem 21

A random sample of 16 tires was tested to estimate the average life of this type of tire under normal driving conditions. The sample mean and sample standard deviation were found to be 47,500 miles and 4,200 miles, respectively.
a. Calculate the margin of error for a $95 \%$ confidence interval estimate of the mean lifetime of this type of tire if driven under normal driving conditions.
b. Find the UCL and the LCL of a $90 \%$ confidence interval estimate of the mean lifetime of this type of tire if driven under normal driving conditions.

Sheryl Ezze

Numerade Educator

Problem 22

Calculate the width for each of the following.
a. $n=6 ; s=40 ; \alpha=0.05$
b. $n=22 ; s^2=400 ; \alpha=0.01$
c. $n=25 ; s=50 ; \alpha=0.10$

Sheryl Ezze

Numerade Educator

Problem 23

In Example 7.5 we calculated a 95\% confidence interval estimate of the Healthy Eating Index-2005 score for a random sample of participants at the time of their first interview. Recall that there are two observations for each person in the study. The first observation, identified by daycode $=1$, contains data from the first interview and the second observation, daycode $=2$, contains data from the second interview. Find a 95\% confidence interval for the mean HEI-2005 score for participants at the time of their second interview. The data are stored in the data file HEI Cost Data Variable Subset.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 24

A machine that packages 18-ounce (510-gram) boxes of sugar-coated wheat cereal is being studied. The weights for a random sample of 100 boxes of cereal packaged by this machine are contained in the data file Sugar.
a. Find a $90 \%$ confidence interval for the population mean cereal weight.
b. Without doing the calculations, state whether an $80 \%$ confidence interval for the population mean would be wider than, narrower than, or the same as the answer to part a.

MA

Melissa A

Numerade Educator

Problem 25

How much do students pay, on the average, for textbooks during the first semester of college? From a random sample of 400 students the mean cost was found to be $\$ 357.75$, and the sample standard deviation was $\$ 37.89$. Assuming that the population is normally distributed, find the margin of error of a $95 \%$ confidence interval for the population mean.

Harsh Gadhiya

Numerade Educator

Problem 26

There is concern about the speed of automobiles traveling over a particular stretch of highway. For a random sample of 28 automobiles, radar indicated the following speeds, in miles per hour:

$$
\begin{array}{lllllll}
59 & 63 & 68 & 57 & 56 & 71 & 59 \\
69 & 53 & 58 & 60 & 66 & 51 & 59 \\
54 & 64 & 58 & 57 & 66 & 61 & 65 \\
70 & 63 & 65 & 57 & 56 & 61 & 59
\end{array}
$$

Assuming a normal population distribution (See Exercise 7.1), find the margin of error of a $95 \%$ confidence interval for the mean speed of all automobiles traveling over this stretch of highway.

Sheryl Ezze

Numerade Educator

Problem 27

A clinic offers a weight-loss program. A review of its records found the following amounts of weight loss, in pounds, for a random sample of 24 of its clients at the conclusion of a 4-month program:

$$
\begin{array}{llllllll}
18 & 25 & 16 & 11 & 15 & 20 & 16 & 19 \\
28 & 25 & 26 & 31 & 45 & 40 & 36 & 19 \\
28 & 25 & 36 & 16 & 35 & 20 & 16 & 19
\end{array}
$$
$$
\begin{array}{llllllll}
18 & 25 & 16 & 11 & 15 & 20 & 16 & 19 \\
28 & 25 & 26 & 31 & 45 & 40 & 36 & 19 \\
28 & 25 & 36 & 16 & 35 & 20 & 16 & 19
\end{array}
$$
would be wider than, narrower than, or the same as that found in part a.

Sheryl Ezze

Numerade Educator

Problem 28

A business school placement director wants to estimate the mean annual salaries 5 years after students graduate. A random sample of 25 such graduates found a sample mean of $$\$ 42,740$$ and a sample standard deviation of $$\$ 4,780$$. Find a $90 \%$ confidence interval for the population mean, assuming that the population distribution is normal.

Sheryl Ezze

Numerade Educator

Problem 29

A car-rental company is interested in the amount of time its vehicles are out of operation for repair work. State all assumptions and find a $90 \%$ confidence interval for the mean number of days in a year that all vehicles in the company's fleet are out of operation if a random sample of nine cars showed the following number of days that each had been inoperative:

$$
\begin{array}{lllllllll}
16 & 10 & 21 & 22 & 8 & 17 & 19 & 14 & 19
\end{array}
$$

Sheryl Ezze

Numerade Educator

Problem 30

Find the margin of error to estimate the population proportion for each of the following.
a. $n=350 ; \hat{p}=0.30 ; \alpha=0.01$
b. $n=275 ; \hat{p}=0.45 ; \alpha=0.05$
c. $n=500 ; \hat{p}=0.05 ; \alpha=0.10$

Sheryl Ezze

Numerade Educator

Problem 31

Calculate the confidence interval to estimate the population proportion for each of the following.
a. $98 \%$ confidence level; $n=450 ; \hat{p}=0.10$
b. $95 \%$ confidence level; $n=240 ; \hat{p}=0.01$
c. $\alpha=0.04 ; n=265 ; \hat{p}=0.50$

Sheryl Ezze

Numerade Educator

Problem 32

A small private university is planning to start a volunteer football program. A random sample of alumni is surveyed. It was found that 250 were in favor of this program, 75 were opposed, and 25 had no opinion.
a. Estimate the percent of alumni in favor of this program. Let $\alpha=0.05$.
b. Estimate the percent of alumni opposed to this volunteer football program with a $90 \%$ confidence level.

Sheryl Ezze

Numerade Educator

Problem 33

Suppose that a random sample of 142 graduate-admissions personnel was asked what role scores on standardized tests (such as the GMAT or GRE) play in the consideration of a candidate for graduate school. Of these sample members, 87 answered "very important." Find a $95 \%$ confidence interval for the population proportion of graduate admissions personnel with this view.

Sheryl Ezze

Numerade Educator

Problem 34

In a random sample of 95 manufacturing firms, 67 indicated that their company attained ISO certification within the last two years. Find a $99 \%$ confidence interval for the population proportion of companies that have been certified within the last 2 years.

Sheryl Ezze

Numerade Educator

Problem 35

The Mendez Mortgage Company case study was given in Chapter 2. A random sample of $n=350$ accounts of the company's total portfolio was selected. Estimate the proportion of all the company's accounts with an original purchase price of less than $$\$ 10,000$$. The data is stored in the data file Mendez Mortgage. Use $\alpha=0.02$.

Rashmi Sinha

Numerade Educator

Problem 36

Consider again the Mendez Mortgage Company case study in Chapter 2. From a random sample of $n=350$ accounts of the company's total portfolio, estimate with $95 \%$ confidence the proportion of all the company's accounts in which the purchaser's latest FICO score was at least 750. The data is stored in the data file Mendez Mortgage.

Rashmi Sinha

Numerade Educator

Problem 37

From a random sample of 400 registered voters in one city, 320 indicated that they would vote in favor of a proposed policy in an upcoming election.
a. Calculate the LCL for a $98 \%$ confidence interval estimate for the population proportion in favor of this policy.
b. Calculate the width of a $90 \%$ confidence interval estimate for the population proportion in favor of this policy.

Sheryl Ezze

Numerade Educator

Problem 38

Of a random sample of 250 marketing students, 180 rated a case of résumé inflation as unethical. Based on this information a statistician computed a confidence interval extending from 0.68 to 0.76 for the population proportion. What is the confidence level of this interval?

Sheryl Ezze

Numerade Educator

Problem 39

A Malaysian airline wanted to determine if customers would be interested in paying a $\$ 10$ flat fee for unlimited Internet access during long-haul flights. From a random sample of 200 customers, 125 indicated that they would be willing to pay this fee. Using this survey data, determine the $99 \%$ confidence interval estimate for the population proportion of the airline's customers who would be prepared to pay this fee for Internet use.

Sheryl Ezze

Numerade Educator

Problem 40

Suppose that the local authorities in a heavily populated residential area of downtown Hong Kong were considering building a new municipal swimming pool and leisure center. Because such a development would cost a great deal of money, it first of all needed to be established whether the residents of this area thought that the swimming pool and leisure center would be a worthwhile use of public funds. If 243 out of a random sample of 360 residents in the local area thought that the pool and leisure center should be built, determine with $95 \%$ confidence the proportion of all the local residents in the area who would support the proposal.

Sheryl Ezze

Numerade Educator

Problem 41

It is important for airlines to follow the published scheduled departure times of flights. Suppose that one airline that recently sampled the records of 246 flights originating in Orlando found that 10 flights were delayed for severe weather, 4 flights were delayed for maintenance concerns, and all the other flights were on time.
a. Estimate the percentage of on-time departures using a $98 \%$ confidence level.
b. Estimate the percentage of flights delayed for severe weather using a $98 \%$ confidence level.

Sheryl Ezze

Numerade Educator

Problem 42

Find the lower confidence limit for the population variance for each of the following normal populations.
a. $n=21 ; \alpha=0.05 ; \mathrm{s}^2=16$
b. $n=16 ; \alpha=0.05 ; s=8$
c. $n=28 ; \alpha=0.01 ; s=15$

Sheryl Ezze

Numerade Educator

Problem 43

Find the upper confidence limit for parts a-c of Exercise 7.42.

Sheryl Ezze

Numerade Educator

Problem 44

Consider the following random sample from a normal population:
$\begin{array}{lllll}12 & 16 & 8 & 10 & 9\end{array}$
a. Find the $90 \%$ confidence interval for population variance.
b. Find the $95 \%$ confidence interval for the population variance.

Sheryl Ezze

Numerade Educator

Problem 46

A clinic offers a weight-loss program. A review of its records found the following amounts of weight loss, in pounds, for a random sample of 10 clients at the conclusion of the program:
$\begin{array}{llllllllll}18.2 & 25.9 & 6.3 & 11.8 & 15.4 & 20.3 & 16.8 & 18.5 & 12.3 & 17.2\end{array}$

James Kiss

Numerade Educator

Problem 47

The quality-control manager of a chemical company randomly sampled twenty 100 -pound bags of fertilizer to estimate the variance in the pounds of impurities. The sample variance was found to be 6.62 . Find a $95 \%$ confidence interval for the population variance in the pounds of impurities.

Sheryl Ezze

Numerade Educator

Problem 48

A psychologist wants to estimate the variance of employee test scores. A random sample of 18 scores had a sample standard deviation of 10.4 . Find a $90 \%$ confidence interval for the population variance. What are the assumptions, if any, to calculate this interval estimate?

Sheryl Ezze

Numerade Educator

Problem 49

A manufacturer is concerned about the variability of the levels of impurity contained in consignments of raw material from a supplier. A random sample of 15 consignments showed a standard deviation of 2.36 in the concentration of impurity levels. Assume normality.
a. Find a $95 \%$ confidence interval for the population variance.
b. Would a $99 \%$ confidence interval for this variance be wider or narrower than that found in part a?

Sheryl Ezze

Numerade Educator

Problem 50

A manufacturer bonds a plastic coating to a metal surface. A random sample of nine observations on the thickness of this coating is taken from a week's output, and the thicknesses (in millimeters) of these observations are as follows:
$\begin{array}{lllllllll}19.8 & 21.2 & 18.6 & 20.4 & 21.6 & 19.8 & 19.9 & 20.3 & 20.8\end{array}$
Assuming normality, find a $90 \%$ confidence interval for the population variance.

Sheryl Ezze

Numerade Educator

Problem 51

Assume simple random sampling. Calculate the variance of the sample mean, $\sigma_x^2$ for each of the following.
a. $N=1200 ; n=80 ; s=10$
b. $N=1425 ; n=90 ; s^2=64$
c. $N=3200 ; n=200 ; s^2=129$

Sheryl Ezze

Numerade Educator

Problem 52

Assume simple random sampling. Calculate the $95 \%$ confidence interval estimate for the population mean for each of the following.
a. $N=1200 ; n=80 ; s=10 ; \bar{x}=142$
b. $N=1425 ; n=90 ; s^2=64 ; \bar{x}=232.4$
c. $N=3200 ; n=200 ; s^2=129 ; \bar{x}=59.3$

Sheryl Ezze

Numerade Educator

Problem 53

Assume simple random sampling. Calculate the confidence interval for the population total for each of the following.
a. $N=1325 ; n=121 ; s=20 ; \bar{x}=182$;
$ 95\%$ confidence level
b. $N=2100 ; n=144 ; s=50 ; \bar{x}=1,325$;
$98 \%$ confidence level

Sheryl Ezze

Numerade Educator

Problem 54

Assume simple random sampling. Calculate the confidence interval for the population proportion, $P$, for each of the following.
a. $N=1058 ; n=160 ; x=40 ; 95 \%$ confidence level
b. $N=854 ; n=81 ; x=50 ; 99 \%$ confidence level

Sheryl Ezze

Numerade Educator

Problem 55

Take a random sample of 50 pages from this book and estimate the proportion of all pages that contain figures.

Narayan Hari

Numerade Educator

Problem 56

A firm employs 189 junior accountants. In a random sample of 50 of these, the mean number of hours overtime billed in a particular week was 9.7, and the sample standard deviation was 6.2 hours.
a. Find a $95 \%$ confidence interval for the mean number of hours overtime billed per junior accountant in this firm that week.
b. Find a $99 \%$ confidence interval for the total number of hours overtime billed by junior accountants in the firm during the week of interest.

Sheryl Ezze

Numerade Educator

Problem 57

An auditor, examining a total of 820 accounts receivable of a corporation, took a random sample of 60 of them. The sample mean was $$\$ 127.43$$, and the sample standard deviation was $$\$ 43.27$$.
a. Using an unbiased estimation procedure, find an estimate of the population mean.
b. Using an unbiased estimation procedure, find an estimate of the variance of the sample mean.
c. Find a $90 \%$ confidence interval for the population mean.
d. A statistician found, for the population mean, a confidence interval running from $$\$ 117.43$$ to $$\$ 137.43$$. What is the probability content of this interval?
e. Find a $95 \%$ confidence interval for the total amount of these 820 accounts.

Sheryl Ezze

Numerade Educator

Problem 58

On a particular day a consumer-advice bureau received 125 calls. For a random sample of 40 of these calls, it was found that mean time taken in providing the requested advice was 7.28 minutes, and the sample standard deviation was 5.32 minutes.
a. Find a $99 \%$ confidence interval for the mean time taken per call.
b. Find a $90 \%$ confidence interval for the total amount of time taken in answering these 125 calls.

Sheryl Ezze

Numerade Educator

Problem 59

State whether each of the following statements is true or false.
a. For a given number of population members and a given sample variance, the larger the number of sample members, the wider the $95 \%$ confidence interval for the population mean.
b. For a given number of population members and a given number of sample members, the larger the sample variance, the wider the $95 \%$ confidence interval for the population mean.
c. For a given number of sample members and a given sample variance, the larger the number of population members, the wider the $95 \%$ confidence interval for the population mean. Justify your answer.
d. For a given number of population members, a given number of sample members, and a given sample variance, a 95\% confidence interval for the population mean is wider than a $90 \%$ confidence interval for the population mean.

Tyler Moulton

Numerade Educator

Problem 60

A senior manager, responsible for a group of 120 junior executives, is interested in the total amount of time per week spent by these people in internal meetings. A random sample of 35 of these executives was asked to keep diary records during the next week. When the results were analyzed, it was found that these sample members spent a total of 143 hours in internal meetings. The sample standard deviation was 3.1 hours. Find a $90 \%$ confidence interval for the total number of hours spent in internal meetings by all 120 junior executives in the week.

Sheryl Ezze

Numerade Educator

Problem 61

A simple random sample of 300 branches out of a total of 1200 branches of a UK travel agency found that 75 had at least one staff member over the age of 55 . Find a $95 \%$ confidence interval for the proportion of all the branches having a staff member over 55 .

Sheryl Ezze

Numerade Educator

Problem 62

A business school dean is contemplating proposing a change in the requirements for graduation. At present, business majors are required to take one science course, chosen from a list of possible courses. The proposal is that this be replaced by the requirement that a course in ecology be taken. The business school has 420 students. In a random sample of 100 of these students, 56 expressed opposition to this proposal. Find a $90 \%$ confidence interval for the proportion of all the school's students opposed to the proposed change in requirements.

Sheryl Ezze

Numerade Educator

Problem 63

An accounting firm has 1200 clients. From a random sample of 120 clients, 110 indicated very high satisfaction with the firm's service. Find a $95 \%$ confidence interval for the proportion of all clients who are very highly satisfied with this firm.

Sheryl Ezze

Numerade Educator

Problem 64

A class has 420 students. The final examination is optional-taking it can raise, but cannot lower, a student's grade. Of a random sample of 80 students, 31 indicated that they would take the final examination. Find a $90 \%$ confidence interval for the total number of students in this class intending to take the final examination.

Sheryl Ezze

Numerade Educator

Problem 65

How large of a sample is needed to estimate the mean of a normally distributed population for each of the following?
a. $M E=5 ; \sigma=40 ; \alpha=0.01$
b. $M E=10 ; \sigma=40 ; \alpha=0.01$
c. Compare and comment on your answers to parts a and $b$.

Sheryl Ezze

Numerade Educator

Problem 66

How large a sample is needed to estimate the population proportion for each of the following?
a. $M E=0.03 ; \alpha=0.05$
b. $M E=0.05 ; \alpha=0.05$
c. Compare and comment on your answers to parts $a$ and $b$.

Sheryl Ezze

Numerade Educator

Problem 67

How large a sample is needed to estimate the population proportion for each of the following?
a. $M E=0.05 ; \alpha=0.01$
b. $M E=0.05 ; \alpha=0.10$
c. Compare and comment on your answers to parts $a$ and $b$.

Sheryl Ezze

Numerade Educator

Problem 68

A research group wants to estimate the proportion of consumers who plan to buy a scanner for their PC during the next 3 months.
a. How many people should be sampled so that the sampling error is at most 0.04 with a $90 \%$ confidence interval?
b. What is the sample size required if the confidence is increased to $95 \%$, keeping the sampling error the same?
c. What is the required sample size if the research group extends the sampling error to 0.05 and wants a $98 \%$ confidence level?

Sheryl Ezze

Numerade Educator

Problem 69

A politician wants to estimate the proportion of constituents favoring a controversial piece of proposed legislation. Suppose that a $99 \%$ confidence interval that extends at most 0.05 on each side of the sample proportion is required. How many sample observations are needed?

Sheryl Ezze

Numerade Educator

Problem 70

The student government association at a university wants to estimate the percentage of the student body that supports a change being considered in the academic calendar of the university for the next academic year. How many students should be surveyed if a $90 \%$ confidence interval is desired and the margin of error is to be only $3 \%$ ?

Sheryl Ezze

Numerade Educator

Problem 71

Determine the sample size needed for each of the following situations.
a. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=50$
b. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=100$
c. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=200$
d. Compare and comment on your answers to parts a through c.

Sheryl Ezze

Numerade Educator

Problem 72

Determine the sample size needed for each of the following situations.
a. $N=3,300$
$\sigma=500$
$1.96 \sigma_{\bar{x}}=50$
b. $N=4,950$
$\sigma=500$
$1.96 \sigma_{\bar{x}}=50$
c. $N=5,000,000 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=50$
d. Compare and comment on your answers to parts a through $c$.

Sheryl Ezze

Numerade Educator

Problem 73

Determine the sample size for each of the following situations.
a. $N=2,500 \quad \hat{p}=0.5 \quad 1.96 \sigma_{\hat{p}}=0.05$
b. $N=2,500 \quad \hat{p}=0.5 \quad 1.96 \sigma_{\hat{p}}=0.03$
c. Compare and comment on your answers to part a and part $b$.

Sheryl Ezze

Numerade Educator

Problem 75

The mean amount of the 812 mortgages taken out in a city in the past year must be estimated. Based on previous experience, a real estate broker knows that the population standard deviation is likely to be about $$\$ 20,000$$. If a $95 \%$ confidence interval for the population mean is to extend $$\$ 2,000$$ on each side of the sample mean, how many sample observations are needed if a simple random sample is taken?

Sheryl Ezze

Numerade Educator

Problem 76

A country club wants to poll a random sample of its 320 members to estimate the proportion likely to attend an early-season function. The number of sample observations should be sufficiently large to ensure that a $99 \%$ confidence interval for the population extends at most 0.05 on each side of the sample proportion. How large of a sample is necessary?

Sheryl Ezze

Numerade Educator

Problem 76

An instructor in a class of 417 students is considering the possibility of a take-home final examination. She wants to take a random sample of class members to estimate the proportion who prefer this form of examination. If a $90 \%$ confidence interval for the population proportion must extend at most 0.04 on each side of the sample proportion, how large a sample is needed?

Sheryl Ezze

Numerade Educator

Problem 77

An instructor in a class of 417 students is considering the possibility of a take-home final examination. She wants to take a random sample of class members to estimate the proportion who prefer this form of examination. If a $90 \%$ confidence interval for the population proportion must extend at most 0.04 on each side of the sample proportion, how large a sample is needed?

Sheryl Ezze

Numerade Educator

Problem 78

Several drugs are used to treat diabetes. A sales specialist for a leading pharmaceutical company needs an estimate of the number of new prescriptions that were written during a particular month for his company's new diabetes drug. The numbers of new prescriptions in a random sample of 25 sales districts are as follows:
$\begin{array}{llllllllll}210 & 240 & 190 & 275 & 290 & 185 & 223 & 190 & 185 & 192 \\ 265 & 312 & 284 & 261 & 243 & 168 & 240 & 170 & 187 & 190 \\ 215 & 240 & 210 & 235 & 290 & & & & & \end{array}$
a. Find a $90 \%$ confidence interval for the average number of new prescriptions written for this new drug among all the sales districts. State the assumptions.
b. Calculate the widths for $95 \%$ and $98 \%$ confidence intervals.

Sheryl Ezze

Numerade Educator

Problem 79

Suppose that the owner of a recently opened convenience store in Kuala Lumpur, Malaysia, wants to estimate how many pounds of bananas are sold during a typical day. The owner checks his sales records for a random sample of 16 days and establishes that the mean number of pounds sold per day is 75 pounds and that the sample standard deviation is 6 pounds. Estimate the mean number of pounds the owner should stock each day to a $95 \%$ confidence level.

Sheryl Ezze

Numerade Educator

Problem 81

The following data represent the number of audience members per week at a theater in Paris during the last year. (The theater was closed for 2 weeks for refurbishment.)
$\begin{array}{llllllllll}163 & 165 & 094 & 137 & 123 & 095 & 170 & 096 & 117 & 129 \\ 152 & 138 & 147 & 119 & 166 & 125 & 148 & 180 & 152 & 149 \\ 167 & 120 & 129 & 159 & 150 & 119 & 113 & 147 & 169 & 151 \\ 116 & 150 & 110 & 110 & 143 & 090 & 134 & 145 & 156 & 165 \\ 174 & 133 & 128 & 100 & 086 & 148 & 139 & 150 & 145 & 100\end{array}$
Estimate the average weekly attendance with a $95 \%$ interval estimate.

Sheryl Ezze

Numerade Educator

Problem 82

The manager of a local fitness center wants an estimate of the number of times members use the weight room per month. From a random sample of 25 members the average number of visits to the weight room over the course of a month was 12.5 visits with a standard deviation of 3.8 visits. Assuming that the monthly number of visits is normally distributed, determine a $95 \%$ confidence interval for the average monthly usage of all members of this fitness center.

Sheryl Ezze

Numerade Educator

Problem 83

Eight randomly selected batches of a chemical were tested for impurity concentration. The percentage impurity levels found in this sample were as follows:
$\begin{array}{llllllll}3.2 & 4.3 & 2.1 & 2.8 & 3.2 & 3.6 & 4.0 & 3.8\end{array}$
a. Find the most efficient estimates of the population mean and variance.
b. Estimate the proportion of batches with impurity levels gre
ater than $3.75 \%$.

Sheryl Ezze

Numerade Educator

Problem 84

A marketing research assistant for a veterinary hospital surveyed a random sample of 457 pet owners. Respondents were asked to indicate the number of times that they visit their veterinarian each year. The sample mean response was 3.59 and the sample standard deviation was 1.045. Based on these results, a confidence interval from 3.49 to 3.69 was calculated for the population mean. Find the probability content for this interval.

Sheryl Ezze

Numerade Educator

Problem 85

A random sample of 174 college students was asked to indicate the number of hours per week that they surf the Internet for either personal information or material for a class assignment. The sample mean response was 6.06 hours and the sample standard deviation was 1.43 hours. Based on these results, a confidence interval extending from 5.96 to 6.16 was calculated for the population mean. Find the confidence level of this interval.

Sheryl Ezze

Numerade Educator

Problem 86

A sample of 33 accounting students recorded the number of hours that they spent studying for a final exam. The data are stored in the data file Study.
a. Give an example of an unbiased, consistent, and efficient estimator of the population mean.
b. Find the sampling error for a $95 \%$ confidence interval estimate of the mean number of hours students studied for this exam.

Sheryl Ezze

Numerade Educator

Problem 87

Dr. Mihaela Sabou wants to estimate the average length of a hospital stay (number of days) for patients with a certain infectious disease. From a random sample of 25 patient records, she finds that the average number of days in the hospital for such patients is 6 days, with a standard deviation of 1.8 days.
a. Find the reliability factor for a $95 \%$ confidence interval estimate of the population mean length of stay.
b. Find the LCL for a $99 \%$ confidence interval estimate of the population mean length of stay.

Sheryl Ezze

Numerade Educator

Problem 88

Suppose that a survey of race fans at this week's Daytona 500 NASCAR race were asked, Is this your first time attending the Daytona 500? From a random sample of 250 race fans, 100 answered in the affirmative.
a. Find the standard error to estimate the population proportion of first timers.
b. Find the sampling error to estimate the population proportion of first timers with $95 \%$ confidence level.
c. Estimate the proportion of repeat fans with $92 \%$ confidence level.

Sheryl Ezze

Numerade Educator

Problem 89

The following data represent the number of passengers per flight in a random sample of 20 flights from Vienna, Austria, to Cluj-Napoca, Romania, with a new airline:
$\begin{array}{llllllllll}63 & 65 & 94 & 37 & 83 & 95 & 70 & 96 & 47 & 29 \\ 52 & 38 & 47 & 79 & 66 & 25 & 48 & 80 & 52 & 49\end{array}$
a. What is the reliability factor for a $90 \%$ confidence interval estimate of the mean number of passengers per flight?
b. Find the LCL for a $99 \%$ confidence interval estimate of the mean number of passengers per flight.

Sheryl Ezze

Numerade Educator

Problem 90

What is the most common method to renew vehicle registration? In checking a random sample of 500 motor vehicle renewal registrations in one county, the finance department found that 200 were mailed, 160 were paid in person at the county finance department office, and the remainder was paid online at the county's Web site. Phone registration renewals were not available.
a. Estimate the population proportion to pay for vehicle registration renewals in person at the county finance department office. Use a $90 \%$ confidence level.
b. Estimate the population proportion of online renewals. Use a 95\% confidence level.

Sheryl Ezze

Numerade Educator

Problem 91

Consider the data in Exercise 7.90. Suppose that we computed for the population proportion who pay for vehicle registration by mail a confidence interval extending from 0.34 to 0.46 . What is the confidence level of this interval?

Sheryl Ezze

Numerade Educator

Problem 92

Consider the data in Exercise 7.90. It was reported in the local paper that less than one-third (from $23.7 \%$ to $32.3 \%$ ) of the population prefers the online renewal process. What is the confidence level of this interval estimate?

Sheryl Ezze

Numerade Educator

Problem 93

The county finance department in Exercise 7.90 also wants information about renewals of disabled parking placards. Suppose that in a sample of 350 transactions for disabled parking placards, it was found that 250 were paid electronically.
a. What is the margin of error for a $99 \%$ confidence interval estimate of the population proportion of disabled renewal transactions paid electronically?
b. Without calculating, is the margin of error for a $95 \%$ confidence interval estimate of the population proportion of disabled renewal transactions paid electronically larger, smaller, or the same as that found in part a for a $99 \%$ confidence interval?

Sheryl Ezze

Numerade Educator

Problem 94

What is the typical age of a person who renews his or her driver's license online? From a random sample of 460 driver's license renewal transactions, the mean age was 42.6 and the standard deviation was 5.4. Compute the $98 \%$ confidence interval estimate of the mean age of online renewal users in this county.

Sheryl Ezze

Numerade Educator

Problem 95

A test was taken by 90 students. A random sample of 10 scores found the following results:
$\begin{array}{llllllllll}93 & 71 & 62 & 75 & 81 & 63 & 87 & 59 & 84 & 72\end{array}$
a. Find a $90 \%$ confidence interval for the population's mean score.
b. Without doing the calculations, state whether a $95 \%$ confidence interval for the population mean would be wider or narrower than the interval found in part a.

Sheryl Ezze

Numerade Educator

Problem 96

A corporation has 272 accounts receivable in a particular category. A random sample of 50 of them was taken. The sample mean was $$\$ 492.36$$, and the sample standard deviation was $$\$ 149.92$$.
a. Find a $99 \%$ confidence interval for the population mean value of these accounts receivable.
b. Find a $95 \%$ confidence interval for the total value of these accounts receivable.
c. Without doing the calculations, state whether a $90 \%$ confidence interval for the population total would be wider or narrower than the interval found in part b.

Sheryl Ezze

Numerade Educator

Problem 97

A corporation employs 148 sales representatives. A random sample of 60 of them was taken, and it was found that, for 36 of the sample members, the volume of orders taken this month was higher than for the same month last year. Find a $95 \%$ confidence interval for the population proportion of sales representatives with a higher volume of orders.

Sheryl Ezze

Numerade Educator

Problem 98

Several drugs are used to treat high blood pressure. A sales specialist for a leading pharmaceutical company randomly sampled the records of 10 sales districts to estimate the number of new prescriptions that had been written during a particular month for the company's new blood pressure medication. The numbers of new prescriptions were as follows:
$$
210,240,190,275,290,265,312,284,261,243
$$
a. Find a $90 \%$ confidence interval for the average number of new prescriptions written for this new drug among all the sales districts. What are the assumptions?
b. Assuming that the confidence level remains constant, what sample size is needed to reduce by half the margin of error of the confidence interval in part a?

Sheryl Ezze

Numerade Educator

Problem 99

The president's policy on domestic affairs received a $45 \%$ approval rating in a recent poll. The margin of error was given as 0.035 . What sample size was used for this poll if we assume a $95 \%$ confidence level?

Sheryl Ezze

Numerade Educator

Problem 100

An automobile dealer has an inventory of 328 used cars. The mean mileage of these vehicles is to be estimated. Previous experience suggests that the population standard deviation is likely to be about 12,000 miles. If a $90 \%$ confidence interval for the population mean is to extend 2,000 miles on each side of the sample mean, how large of a sample is required if simple random sampling is employed?

Sheryl Ezze

Numerade Educator

Problem 101

A simple random sample is to be taken of 527 business majors in a college to estimate the proportion favoring greater emphasis on business ethics in the curriculum. How many observations are necessary to ensure that a $95 \%$ confidence interval for the population proportion extends at most 0.06 on each side of the sample proportion?

Sheryl Ezze

Numerade Educator

Problem 102

Show algebraically that Equation 7.23 is equal to Equation 7.24. That is,
$$
\frac{N \sigma^2}{(N-1) \sigma_{\bar{X}}^2+\sigma^2}=\frac{n_0 N}{n_0+(N-1)}
$$

Sheryl Ezze

Numerade Educator

Problem 103

The demand for bottled water increases during the hurricane season in Florida. The operations manager at a plant that bottles drinking water
wants to be sure that the filling process for 1-gallon bottles (1 gallon is approximately 3.785 liters) is operating properly. Currently, the company is testing the volumes of 1-gallon bottles. Suppose that a random sample of 75 one-gallon bottles is tested. Find the $95 \%$ confidence interval estimate of the population mean volume. The measurements are recorded in the data file Water.

Harsh Gadhiya

Numerade Educator

Problem 803

Everyone knows that exercise is important. Recently, employees of one large international corporation were surveyed and asked, How many minutes do you spend daily on some form of rigorous exercise? From a random sample of 25 employees, the mean time spent on vigorous daily exercise was 28.5 minutes. The standard deviation was found to be 6.8 minutes. Find a $90 \%$ interval estimate of the mean daily time spent on rigorous exercise by all employees.

Sheryl Ezze

Numerade Educator