Introduction to Probability and Statistics

William Mendenhall, III, Robert J. Beaver, Barbara M. Beaver

Chapter 8

Large-Sample Estimation - all with Video Answers

Educators

Section 2

Point Estimation

Problem 1

Explain what is meant by "margin of error" in point estimation.

Sheryl Ezze

Sheryl Ezze

Numerade Educator

Problem 2

What are two characteristics of the best point estimator for a population parameter?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 3

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises 3-6. Comment on how a larger population variance affects the margin of error:
$n=30, \sigma^{2}=.2$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 4

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises 3-6. Comment on how a larger population variance affects the margin of error:
$n=30, \sigma^{2}=.9$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 5

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises 3-6. Comment on how a larger population variance affects the margin of error:
$n=30, \sigma^{2}=1.5$

Harsh Gadhiya

Numerade Educator

Problem 6

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises 3-6. Comment on how a larger population variance affects the margin of error:
$n=30, \sigma^{2}=3.8$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 7

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises $7-10 .$ Comment on how an increased sample size affects the margin of error:
$n=50, s^{2}=4$

Sheryl Ezze

Numerade Educator

Problem 8

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises $7-10 .$ Comment on how an increased sample size affects the margin of error:
$n=100, s^{2}=4$

Lucas Finney

Numerade Educator

Problem 9

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises $7-10 .$ Comment on how an increased sample size affects the margin of error:
$n=500, s^{2}=4$

Sheryl Ezze

Numerade Educator

Problem 10

Calculate the margin of error in estimating a population mean $\mu$ for the values given in Exercises $7-10 .$ Comment on how an increased sample size affects the margin of error:
$n=1000, s^{2}=4$

Lucas Finney

Numerade Educator

Problem 11

Calculate the margin of error in estimating a binomial proportion $p$ for the sample sizes given in Exercises $11-14$. Use $p=.5$ to calculate the standard error of the estimator, and comment on how an increased sample size affects the margin of error.
$n=30$

Michael Anderson

Michael Anderson

Numerade Educator

Problem 12

Calculate the margin of error in estimating a binomial proportion $p$ for the sample sizes given in Exercises $11-14$. Use $p=.5$ to calculate the standard error of the estimator, and comment on how an increased sample size affects the margin of error.
$n=100$

Michael Anderson

Michael Anderson

Numerade Educator

Problem 13

Calculate the margin of error in estimating a binomial proportion $p$ for the sample sizes given in Exercises $11-14$. Use $p=.5$ to calculate the standard error of the estimator, and comment on how an increased sample size affects the margin of error.
$n=400$

Michael Anderson

Michael Anderson

Numerade Educator

Problem 14

Calculate the margin of error in estimating a binomial proportion $p$ for the sample sizes given in Exercises $11-14$. Use $p=.5$ to calculate the standard error of the estimator, and comment on how an increased sample size affects the margin of error.
$n=1000$

Michael Anderson

Michael Anderson

Numerade Educator

Problem 15

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.1$

Harsh Gadhiya

Numerade Educator

Problem 16

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.3$

Derrick Hanson

Numerade Educator

Problem 17

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.5$

Sheryl Ezze

Numerade Educator

Problem 18

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.7$

Harsh Gadhiya

Numerade Educator

Problem 19

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.9$

Harsh Gadhiya

Numerade Educator

Problem 20

Using the sample information given in Exercises $20-21$, give the best point estimate for the population mean $\mu$ and calculate the margin of error:
A random sample of $n=50$ observations from a quantitative population produced $\bar{x}=56.4$ and $s^{2}=2.6$.

Lucas Finney

Numerade Educator

Problem 21

Using the sample information given in Exercises $20-21$, give the best point estimate for the population mean $\mu$ and calculate the margin of error:
A random sample of $n=75$ observations from a quantitative population produced $\bar{x}=29.7$ and $s^{2}=10.8$.

Lucas Finney

Numerade Educator

Problem 22

Using the sample information given in Exercises $22-23,$ give the best point estimate for the binomial proportion $p$ and calculate the margin of error.
A random sample of $n=900$ observations from a binomial population produced $x=655$ successes.

Derrick Hanson

Numerade Educator

Problem 23

Using the sample information given in Exercises $22-23,$ give the best point estimate for the binomial proportion $p$ and calculate the margin of error.
A random sample of $n=500$ observations from a binomial population produced $x=450$ successes.

Derrick Hanson

Numerade Educator

Problem 24

Using the sample information given in Exercises $22-23,$ give the best point estimate for the binomial proportion $p$ and calculate the margin of error.
Suppose you are writing a questionnaire for a sample survey involving $n=100$ individuals, which will generate estimates for several different binomial proportions. If you want to report a single margin of error for the survey, what margin of error should you use?

Derrick Hanson

Numerade Educator

Problem 25

You want to estimate the mean hourly yield for a process that manufactures an antibiotic. You observe the process for 100 hourly periods chosen at random, with the results $\bar{x}=1020$ grams per hour and $s=90 .$ Estimate the mean hourly yield for the process and calculate the margin of error.

Nick Johnson

Numerade Educator

Problem 26

An experimental rehabilitation technique was used on released convicts. It was shown that 79 of 121 men subjected to the technique pursued useful and crime-free lives for a 3-year period following prison release. Find a point estimate for $p,$ the probability that a convict subjected to the rehabilitation technique will follow a crime-free existence for at least 3 years after prison release. Calculate the margin of error.

Ahmed Genedy

Numerade Educator

Problem 27

If 36 measurements of the specific gravity of aluminum had a mean of 2.705 and a standard deviation of .028 , find the point estimate for the actual specific gravity of aluminum and calculate the margin of error.

Sarah X

Numerade Educator

Problem 28

A geologist studying the movement of the earth's crust at a particular location on California's San Andreas fault found many fractures in the local rock structure. In an attempt to determine the mean angle of the breaks, she sampled $n=50$ fractures and found the sample mean and standard deviation to be $39.8^{\circ}$ and $17.2^{\circ},$ respectively. Estimate the mean angular direction of the fractures and find the margin of error for your estimate.

James Kiss

Numerade Educator

Problem 29

Biomass, the total amount of vegetation held by the earth's forests, is important in determining the amount of unabsorbed carbon dioxide that is expected to remain in the earth's atmosphere. $^{2}$ Suppose a sample of 75 one-square-meter plots, randomly chosen in North America's boreal (northern) forests, produced a mean biomass of 4.2 kilograms per square meter $\left(\mathrm{kg} / \mathrm{m}^{2}\right)$, with a standard deviation of $1.5 \mathrm{~kg} / \mathrm{m}^{2}$ Estimate the average biomass for the boreal forests of North America and find the margin of error for your estimate.

Evelyn Cunningham

Evelyn Cunningham

Numerade Educator

Problem 30

Confidence An increase in the rate of consumer savings is frequently tied to a lack of confidence in the economy. A random sample of $n=200$ savings accounts in a local community showed a mean increase in savings account values of $7.2 \%$ over the past 12 months, with a standard deviation of 5.6\%. Estimate the mean percent increase in savings account values over the past 12 months for depositors in the community. Find the margin of error for your estimate.

Victor Salazar

Numerade Educator

Problem 31

Do our children spend enough time enjoying the outdoors and playing with friends, or are they spending more time glued to the television, computer, and their cell phones? A random sample of 250 youth between the ages of 8 and 18 showed that 170 of them had a TV in their bedroom and that 120 had a video game console in their bedroom.
a. Estimate the proportion of all 8 - to 18 -year-olds who have a TV in their bedroom, and calculate the margin of error for your estimate.
b. Estimate the proportion of all 8 - to 18 -year-olds who have a video game console in their bedroom, and calculate the margin of error for your estimate.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 32

Hotel costs during the summer months can vary substantially depending on the type of room and the amenities offered. ${ }^{3}$ Suppose that we randomly select 50 billing statements from each of the computer databases of the Marriott, Westin, and the Doubletree hotel chains, and record the nightly room rates.
$$\begin{array}{lccc}\hline & \text { Marriott } & \text { Westin } & \text { Doubletree } \\\hline \text { Sample Average (\$) } & 150 & 165 & 125 \\\text { Sample Standard Deviation } & 17.2 & 22.5 & 12.8 \\\hline\end{array}$$
a. Describe the sampled population(s).
b. Find a point estimate for the average room rate for the Marriott hotel chain. Calculate the margin of error. c. Find a point estimate for the average room rate for the Westin hotel chain. Calculate the margin of
error.
d. Find a point estimate for the average room rate for the Doubletree hotel chain. Calculate the margin of
error.
e. Display the results of parts $\mathrm{b}, \mathrm{c},$ and d graphically, using the form shown in Figure 8.5 . Use this display to compare the average room rates for the three hotel chains.

Nick Johnson

Numerade Educator

Problem 33

Radio and television stations often air controversial issues during broadcast time. A poll is then conducted, asking viewers who agree with the issue to call a certain 900 telephone number and those who disagree to call a second 900 telephone number. All respondents pay a fee for their calls.
a. Does this polling technique result in a random sample?
b. What can be said about the validity of the results of such a survey? Do you need to worry about a margin of error in this case?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 34

Do you think that the United States should pursue a program to send humans to Mars? An opinion poll conducted by the Associated Press indicated that $49 \%$ of the 1034 adults surveyed think that we should pursue such a program.
a. Estimate the true proportion of Americans who think that the United States should pursue a program to send humans to Mars. Calculate the margin of error.
b. The question posed in part a was only one of many questions concerning our space program that were asked in the opinion poll. If the Associated Press wanted to report one sampling error that would be valid for the entire poll, what value should they report?

Victor Salazar

Numerade Educator

Problem 35

In an experiment to assess the strength of the hunger drive in rats, 30 previously trained animals were deprived of food for 24 hours. At the end of the 24 -hour period, each animal was put into a cage where food was dispensed if the animal pressed a lever. The length of time the animal continued pressing the bar (although receiving no food) was recorded for each animal. If the data yielded a sample mean of 19.3 minutes with a standard deviation of 5.2 minutes, estimate the true mean time and calculate the margin of error.

Joshua Argo

Numerade Educator