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Practice Makes Perfect Statistics

Sandra McCune

Chapter 10

Estimation - all with Video Answers

Educators

Chapter Questions

01:25

Problem 1

For 1-5, describe the sampling distribution of $\bar{X}$ in terms of its shape, mean, and standard deviation.
A random sample of size $n=100$ is obtained from a population with $\mu=300$ and $\sigma=50$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:52

Problem 1

For 1-3, suppose you have obtained a random sample of $n=15$ measurements from a normally distributed population. Compare $t_{w / 2}$ with $z_{\alpha / 2}$ if you were to form a confidence interval at the indicated level of confidence.
$90 \%$ confidence interval

Jameson Kuper
Jameson Kuper
Numerade Educator
01:52

Problem 1

For $1-5$, for the given value of $\hat{p}$, determine whether the stated sample size $n$ is large enough (both $n \hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 5 ) to use the methods of this section to construct a confidence interval for $p$.
$n=200, \hat{p}=0.2$

Lucas Finney
Lucas Finney
Numerade Educator
01:06

Problem 1

For 1-5, a random sample of $n$ measurements was selected from a population with unknown mean $\mu$ and standard deviation $\sigma$. Construct a $95 \%$ confidence interval for $\mu$ from the information given. Round to two decimal places when needed.
$n=36, \bar{x}=45, s=4.2$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:06

Problem 1

For 1-5, find the sample size necessary to estimate the population mean $\mu$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $\sigma$ is the given value.
$E=2, \sigma=10$

Emily Himsel
Emily Himsel
Numerade Educator
01:48

Problem 2

Describe the sampling distribution of $\bar{X}$ in terms of its shape, mean, and standard deviation.
A random sample of size $n=35$ is obtained from a population with $\mu=60$ and $\sigma=10$.

Jon Southam
Jon Southam
Numerade Educator
01:59

Problem 2

Suppose you have obtained a random sample of $n=15$ measurements from a normally distributed population. Compare $t_{w / 2}$ with $z_{\alpha / 2}$ if you were to form a confidence interval at the indicated level of confidence.
$95 \%$ confidence interval

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:52

Problem 2

For the given value of $\hat{p}$, determine whether the stated sample size $n$ is large enough (both $n \hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 5 ) to use the methods of this section to construct a confidence interval for $p$.
$n=20, \hat{p}=0.2$

Lucas Finney
Lucas Finney
Numerade Educator
02:53

Problem 2

A random sample of $n$ measurements was selected from a population with unknown mean $\mu$ and standard deviation $\sigma$. Construct a $95 \%$ confidence interval for $\mu$ from the information given. Round to two decimal places when needed.
$n=50, \bar{x}=1620, s=215$

Jen H
Jen H
Numerade Educator
02:43

Problem 2

Find the sample size necessary to estimate the population mean $\mu$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $\sigma$ is the given value.
$E=2, \sigma=100$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:52

Problem 3

For the given value of $\hat{p}$, determine whether the stated sample size $n$ is large enough (both $n \hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 5 ) to use the methods of this section to construct a confidence interval for $p$.
$n=100, \hat{p}=0.03$

Lucas Finney
Lucas Finney
Numerade Educator
02:43

Problem 3

Find the sample size necessary to estimate the population mean $\mu$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $\sigma$ is the given value.
$E=0.2, \sigma=100$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:44

Problem 3

Describe the sampling distribution of $\bar{X}$ in terms of its shape, mean, and standard deviation.
A random sample of size $n=9$ is obtained from a normally distributed population with $\mu=25$ and $\sigma=6$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
View

Problem 3

Suppose you have obtained a random sample of $n=15$ measurements from a normally distributed population. Compare $t_{w / 2}$ with $z_{\alpha / 2}$ if you were to form a confidence interval at the indicated level of confidence.
$99 \%$ confidence interval

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:53

Problem 3

A random sample of $n$ measurements was selected from a population with unknown mean $\mu$ and standard deviation $\sigma$. Construct a $95 \%$ confidence interval for $\mu$ from the information given. Round to two decimal places when needed.
$n=200, \bar{x}=105, s=24$

Jen H
Jen H
Numerade Educator
01:29

Problem 4

Find the sample size necessary to estimate the population mean $\mu$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $\sigma$ is the given value.
$E=1.5, \sigma=5$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:52

Problem 4

For the given value of $\hat{p}$, determine whether the stated sample size $n$ is large enough (both $n \hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 5 ) to use the methods of this section to construct a confidence interval for $p$.
$n=15, \hat{p}=0.6$

Lucas Finney
Lucas Finney
Numerade Educator
01:27

Problem 4

Describe the sampling distribution of $\bar{X}$ in terms of its shape, mean, and standard deviation.
A random sample of size $n=36$ is obtained from a population with $\mu=30$ and $\sigma=12$.

Ariana Nash
Ariana Nash
Numerade Educator
02:49

Problem 4

A random sample of $n$ measurements was selected from a population with unknown mean $\mu$ and standard deviation $\sigma$. Construct a $95 \%$ confidence interval for $\mu$ from the information given. Round to two decimal places when needed.
$n=100, \bar{x}=7.8, s=0.5$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:06

Problem 4

For 4-10, a random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$80 \%$ confidence interval, $n=16$

Tyler Moulton
Tyler Moulton
Numerade Educator
02:53

Problem 5

A random sample of $n$ measurements was selected from a population with unknown mean $\mu$ and standard deviation $\sigma$. Construct a $95 \%$ confidence interval for $\mu$ from the information given. Round to two decimal places when needed.
$n=75, \bar{x}=412, s=16$

Jen H
Jen H
Numerade Educator
02:53

Problem 5

A random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$90 \%$ confidence interval, $n=9$

Jen H
Jen H
Numerade Educator
01:27

Problem 5

Describe the sampling distribution of $\bar{X}$ in terms of its shape, mean, and standard deviation.
A random sample of size $n=225$ is obtained from a population with $\mu=120$ and $\sigma=27$.

Ariana Nash
Ariana Nash
Numerade Educator
01:29

Problem 5

Find the sample size necessary to estimate the population mean $\mu$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $\sigma$ is the given value.
$E=0.15, \sigma=5$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:52

Problem 5

For the given value of $\hat{p}$, determine whether the stated sample size $n$ is large enough (both $n \hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 5 ) to use the methods of this section to construct a confidence interval for $p$.
$n=150, \hat{p}=0.01$

Lucas Finney
Lucas Finney
Numerade Educator
01:30

Problem 6

For 6-15, find the indicated probabilities.
A random sample of size $n=100$ is obtained from a population with $\mu=300$ and $\sigma=50$. What is the probability that the sample mean will fall between 299 and 301 ?

Hoan Nguyen
Hoan Nguyen
Numerade Educator
02:53

Problem 6

A random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$90 \%$ confidence interval, $n=25$

Jen H
Jen H
Numerade Educator
01:08

Problem 6

For 6-10, use the sample data and confidence level to construct the confidence interval estimate of the population proportion $p$.
$n=200, \hat{p}=0.2,90 \%$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:53

Problem 6

A random sample of size $n=50$ obtained from a population with unknown mean yielded $\bar{x}=25$. Assume the standard deviation $\sigma=8$. Construct a $95 \%$ confidence interval for $\mu$.

Jen H
Jen H
Numerade Educator
03:02

Problem 6

For 6-10, find the sample size necessary to estimate the population proportion $p$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $p$ has the given value.
$E=0.02, p=0.1$

Jen H
Jen H
Numerade Educator
01:06

Problem 7

Use the sample data and confidence level to construct the confidence interval estimate of the population proportion $p$.
$n=100, \hat{p}=0.45,99 \%$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:53

Problem 7

A random sample of size $n=9$ obtained from a normally distributed population with unknown mean $\mu$ yielded $\bar{x}=26.7$. Assume the standard deviation $\sigma=6$. Construct a $90 \%$ confidence interval for $\mu$.

Jen H
Jen H
Numerade Educator
03:31

Problem 7

Find the indicated probabilities.
A random sample of size $n=36$ is obtained from a population with $\mu=60$ and $\sigma=15$. What is the probability that the sample mean will be at least 58 ?

Wei Zhang
Wei Zhang
Numerade Educator
01:01

Problem 7

Find the sample size necessary to estimate the population proportion $p$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $p$ has the given value.
$E=0.02, p=0.2$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:53

Problem 7

A random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$95 \%$ confidence interval, $n=4$

Jen H
Jen H
Numerade Educator
01:06

Problem 8

Use the sample data and confidence level to construct the confidence interval estimate of the population proportion $p$.
$n=1500, \hat{p}=0.03,95 \%$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:01

Problem 8

Find the sample size necessary to estimate the population proportion $p$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $p$ has the given value.
$E=0.02, p=0.5$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:53

Problem 8

A random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$95 \%$ confidence interval, $n=25$

Jen H
Jen H
Numerade Educator
01:44

Problem 8

Find the indicated probabilities.
A random sample of size $n=9$ is obtained from a normally distributed population with $\mu=25$ and $\sigma=6$. What is the probability that the sample mean will exceed 26.7?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:53

Problem 8

A random sample of size $n=100$ obtained from a population with unknown mean $\mu$ and standard deviation $\sigma$ yielded $\bar{x}=301$ and $s=37$. Construct a $90 \%$ confidence interval for $\mu$.

Jen H
Jen H
Numerade Educator
02:53

Problem 9

A random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$98 \%$ confidence interval, $n=25$

Jen H
Jen H
Numerade Educator
01:01

Problem 9

Find the sample size necessary to estimate the population proportion $p$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $p$ has the given value.
$E=0.02, p=0.7$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:31

Problem 9

Find the indicated probabilities.
A random sample of size $n=36$ is obtained from a population with $\mu=30$ and $\sigma=12$. Find the probability that the sample mean will be no more than 26.08 .

Wei Zhang
Wei Zhang
Numerade Educator
01:03

Problem 9

Use the sample data and confidence level to construct the confidence interval estimate of the population proportion $p$.
$n=15, \hat{p}=0.6,90 \%$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:53

Problem 9

A random sample of size $n=36$ obtained from a population with unknown mean $\mu$ and standard deviation $\sigma$ yielded $\bar{x}=58$ and $s=15$. Construct a $99 \%$ confidence interval for $\mu$.

Jen H
Jen H
Numerade Educator
02:53

Problem 10

A random sample of n measurements obtained from a normally distributed population yielded $\bar{x}=450$ and $s=60$. Construct $a$ confidence interval for $\mu$ at the indicated level of confidence.
$99 \%$ confidence interval, $n=25$

Jen H
Jen H
Numerade Educator
01:03

Problem 10

Use the sample data and confidence level to construct the confidence interval estimate of the population proportion $p$.
$n=1000, \hat{p}=0.82,95 \%$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:01

Problem 10

Find the sample size necessary to estimate the population proportion $p$ to within the indicated margin of error $E$ with $95 \%$ confidence, given that prior data suggest that $p$ has the given value.
$E=0.02, p=0.9$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:15

Problem 10

Find the indicated probabilities.
A random sample of size $n=225$ is obtained from a population with $\mu=120$ and $\sigma=27$. Find the probability that the sample mean will lie between 122 and 124.

Neel Faucher
Neel Faucher
Numerade Educator
02:53

Problem 10

A random sample of size $n=225$ obtained from a population with unknown mean $\mu$ and standard deviation $\sigma$ yielded $\bar{x}=123$ and $s=25$. Construct a $90 \%$ confidence interval for $\mu$.

Jen H
Jen H
Numerade Educator
01:07

Problem 11

Find the indicated probabilities.
A random sample of size $n=25$ is obtained from a normally distributed population with $\mu=120$ and $\sigma=15$. Find the probability that the sample mean will be no more than 126 .

Neel Faucher
Neel Faucher
Numerade Educator
02:44

Problem 11

A random check of 500 batteries manufactured by a company found 15 defective batteries.
(a) Construct a $95 \%$ confidence interval for the proportion of all batteries manufactured by the company that are defective.
(b) Give the margin of error $E$ for the $95 \%$ confidence interval.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:05

Problem 11

A health science center researcher wants to estimate the average amount of fat grams in the signature sandwich of a local deli. A random sample of 40 signature sandwiches yielded $\bar{x}=17$ fat grams and $s=2.5$ fat grams. Construct a $95 \%$ confidence for the average amount of fat grams in the signature sandwich of the local deli.

Lucas Finney
Lucas Finney
Numerade Educator
02:54

Problem 11

A health professional wants to estimate the birth weights of infants. A previous study indicates that the standard deviation of infant birth weights is 7.6 ounces. What size sample is necessary if the health professional wants to estimate the true mean birth weights of infants to within 1.5 ounces with $99 \%$ confidence?

Joshua Argo
Joshua Argo
Numerade Educator
00:59

Problem 11

Suppose that scores on a standardized exam are normally distributed. To estimate the average score on the exam for all test takers, a researcher obtains a random sample of 25 scores. The sample yielded $\bar{x}=480$ and $s=75$. Give a $95 \%$ confidence interval for the average score on the exam for all test takers.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:08

Problem 12

Assume a study revealed that from a sample of 80 working women in a metropolitan area, 12 were over the age of 60 .
(a) Construct a $99 \%$ confidence interval for the true proportion of working women over the age of 60 in the metropolitan area.
(b) Give the margin of error $E$ for the $99 \%$ confidence interval.

Tyler Moulton
Tyler Moulton
Numerade Educator
00:59

Problem 12

A pollster wants to conduct a random survey to estimate the proportion of U.S. citizens who favor a certain Presidential candidate within a margin of error $E=0.04$ ("4 percentage points") with $95 \%$ confidence. If no prior information is available, find the minimum required sample size for the pollster's survey.

Nick Johnson
Nick Johnson
Numerade Educator
01:04

Problem 12

Find the indicated probabilities.
A random sample of size $n=100$ is obtained from a population with $\mu=1200$ and $\sigma=250$. Find the probability that the sample mean will lie between 1200 and 1250.

Neel Faucher
Neel Faucher
Numerade Educator
01:18

Problem 12

The fundraising officer for a charity organization wants to estimate the average donation from contributors to the charity. A random sample of 100 donations yielded $$\bar{x}=\$ 234.85$$ and $$s=\$ 95.23$$. Construct a $90 \%$ confidence for the average donation from all contributors to the charity.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:44

Problem 12

A local deli chain advertises a "heart healthy" sandwich on the menu. Suppose that the amount of fat grams in a "heart healthy" sandwich is a normal random variable. To estimate the average amount of fat grams in "heart healthy" sandwiches, a nutritionist selects a random sample of four "heart healthy" sandwiches. The sample yielded $\bar{x}=11$ grams and $s=1.8$ grams. Construct a $90 \%$ confidence interval for the average amount of fat grams in a "heart healthy" sandwich.

Lucas Finney
Lucas Finney
Numerade Educator
01:46

Problem 13

Find the indicated probabilities.
IQ scores for adults are normally distributed, with mean $\mu=100$ and standard deviation $\sigma=15$. If a random sample of 16 adults is selected, find the probability that the average IQ score in the sample will be at least 92.5 .

Abhishek Kumar
Abhishek Kumar
Numerade Educator
04:15

Problem 13

A radio station advertising manager needs to know the proportion of radio listeners in the area who listen to the manager's radio station. Suppose that a random sample of 100 radio listeners in the area revealed that 34 listen to the manager's station.
(a) Construct a $90 \%$ confidence interval for the proportion of radio listeners in the area who listen to the manager's radio station.
(b) Give the margin of error $E$ for the $90 \%$ confidence interval.

Sanchit Jain
Sanchit Jain
Numerade Educator
02:50

Problem 13

A manufacturer of car batteries needs to estimate the average life of its batteries. A random sample of 80 batteries had $\bar{x}=39$ months and $s=8$ months. Calculate a $95 \%$ confidence interval for the average life of the manufacturer's batteries.

James Kiss
James Kiss
Numerade Educator
02:59

Problem 13

A manufacturer of car batteries needs to estimate the average life of its batteries. Prior data indicate that the standard deviation of the population of batteries is approximately eight months. What size sample is necessary if the manufacturer wants to estimate the true average life of its car batteries within two months with $90 \%$ confidence?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:39

Problem 13

A plumbing company has found that the length of time, in minutes, for installing a bathtub is normally distributed. The owner of the company hires a statistician to estimate the average length of time for installing a bathtub by his company. The statistician randomly selects a sample of 10 bathtub installations. The sample yielded $\bar{x}=150$ minutes and $s=30$ minutes. Construct a $95 \%$ confidence interval for the average length of time for installing a bathtub by the company.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
05:47

Problem 14

An interior designer needs to know the proportion of assisted living residents who prefer a blue hue on bedroom walls. A random survey of 300 assisted living residents revealed that 240 prefer a blue hue on bedroom walls.
(a) Construct a $95 \%$ confidence interval for the proportion of all assisted living residents who prefer a blue hue on bedroom walls.
(b) Give the margin of error $E$ for the $95 \%$ confidence interval.

Willis James
Willis James
Numerade Educator
02:36

Problem 14

The fundraising officer for a charity organization wants to estimate the average donation from contributors to the charity. Prior data indicate that the standard deviation of charitable contributions to the organization is approximately $$\$ 95$$. What size sample is necessary if the fundraising officer wants to estimate the true average donation from all contributors to the charity within $$\$ 20$$ with $95 \%$ confidence?

Norman Atentar
Norman Atentar
Numerade Educator
04:20

Problem 14

Find the indicated probabilities.
Suppose that scores on a national exam are normally distributed, with mean $\mu=500$ and standard deviation $\sigma=100$. If a random sample of 200 test takers is selected, find the probability that the average score of the sample will be between 500 and 520 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:44

Problem 14

A study of 500 randomly selected adults in the United States yielded an average of 2400 calories consumed per day, with a standard deviation of 930 calories. Calculate a $95 \%$ confidence for the true mean caloric intake of all adults in the United States.

Lucas Finney
Lucas Finney
Numerade Educator
01:30

Problem 14

The weights of newborn baby girls born at a local hospital are normally distributed. A hospital administrator conducts a study to estimate the mean weight of all newborn baby girls born at the local hospital. A random sample of size $n=20$ had a mean weight of 95 ounces and a standard deviation of 8 ounces. Calculate a $99 \%$ confidence interval for the mean weight of all newborn baby girls born at the local hospital.

Janet Meinke
Janet Meinke
Numerade Educator
06:18

Problem 15

A quality control inspector for a company that makes cell phones needs to estimate the proportion of defective cell phones produced by the company. The quality control inspector estimates that the proportion defective is about 0.1 , corresponding to $10 \%$ defective. How many of the company's cell phones should be sampled and checked in order to estimate the proportion of defective cell phones to within 0.01 with $90 \%$ confidence?

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:34

Problem 15

The lifetime for a certain brand of incandescent light bulbs is normally distributed. A random sample of five of these light bulbs had a mean lifetime of 520 hours and standard deviation of 50 hours. Give a $95 \%$ confidence interval for the mean lifetime of all incandescent light bulbs of this brand.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 15

A realtor needs to know the average price of a house in a metropolitan area. A random sample of 50 house prices in the metropolitan area yielded $$\bar{x}=\$ 120,000$$ and $$s=\$ 47,000$$. Give a $95 \%$ confidence interval for the average price of a house in the metropolitan area.

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00:55

Problem 15

Find the indicated probabilities.
A local deli chain makes a signature sandwich. Suppose that the amount of fat grams in the signature sandwich are normally distributed, with mean $\mu=16 \mathrm{grams}$ and standard deviation $\sigma=1.5$ grams. If a random sample of four signature sandwiches is selected, find the probability that the average amount of fat grams will be no more than 17 fat grams.

Christopher Stanley
Christopher Stanley
Numerade Educator
03:02

Problem 15

A market researcher wants to know the percent of households in a rural area that have no vehicle. In a random sample of 120 households, 6 households had no vehicle.
(a) Construct a $90 \%$ confidence interval for the percent of households in the rural area that have no vehicle.
(b) Give the margin of error $E$ for the $90 \%$ confidence interval.

Jen H
Jen H
Numerade Educator