Chapter 12, Asking and Answering Questions About a Population Mean Video Solutions, Statistics Learning From Data

Problem 1

A random sample is selected from a population with mean $\mu=100$ and standard deviation $\sigma=10 .$ Determine the mean and standard deviation of the sampling distribution of $\bar{x}$ for each of the following sample sizes:
a. $n=9$
d. $n=50$
b. $n=15$
e. $n=100$
c. $n=36$
f. $n=400$

Prabhakar Kumar

Numerade Educator

00:49

Problem 2

For which of the sample sizes given in the previous exercise would it be reasonable to think that the $\bar{x}$ sampling distribution is approximately normal in shape?

Hossam Mohamed

Numerade Educator

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Problem 3

The paper "Alcohol Consumption, Sleep, and Academic Performance Among College Students" (Journal of Studies on Alcohol and Drugs [2009]: 355-363) describes a study of $n=236$ students who were randomly selected from a list of students enrolled at a liberal arts college in the northeastern region of the United States. Each student in the sample responded to a number of questions about their sleep patterns. For these 236 students, the sample mean time spent sleeping per night was reported to be 7.71 hours and the sample standard deviation of the sleeping times was 1.03 hours. Suppose that you are interested in learning about the value of $\mu,$ the population mean time spent sleeping per night for students at this college. The following table is similar to the table that appears in Example 12.4 . The "what you know" information has been provided. Complete the table by filling in the "how you know it" column.

Rashmi Sinha

Numerade Educator

01:12

Problem 4

Explain the difference between $\mu$ and $\mu_{\vec{x}}$.

Carly Stoner

Numerade Educator

04:09

Problem 5

The time that people have to wait for an elevator in an office building has a uniform distribution over the interval from 0 to 1 minute. For this distribution, $\mu=0.5$ and $\sigma=0.289$
a. If $\bar{x}$ is the average waiting time for a random sample of $n=16$ waiting times, what are the values of the mean and standard deviation of the sampling distribution of $\bar{x} ?$
b. Answer Part (a) for a random sample of 50 waiting times. Draw a picture of the approximate sampling distribution of $\bar{x}$ when $n=50$.

Amany Waheeb

Numerade Educator

03:26

Problem 6

A random sample is selected from a population with mean $\mu=60$ and standard deviation $\sigma=3$. Determine the mean and standard deviation of the sampling distribution of $\bar{x}$ for each of the following sample sizes:
a. $n=6$
d. $n=75$
b. $n=18$
e. $n=200$
c. $n=42$
f. $n=400$

Idabelle Cunningham

Numerade Educator

03:47

Problem 7

12.7 approximately normal in shape?

Marcella Sippey

Numerade Educator

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Problem 8

The paper "Alcohol Consumption, Sleep, and Academic Performance Among College Students" (Journal of Studies on Alcohol and Drugs [2009]: 355-363) describes a study of $n=236$ students that were randomly selected from a list of students enrolled at a liberal arts college in the northeastern region of the United States. Each student in the sample responded to a number of questions about their sleep patterns. For these 236 students, the sample mean additional time spent sleeping on weekend days compared to the other days of the week was reported to be 1.29 hours and the standard deviation was 1.09 hours. Suppose that you are interested in learning about the value of $\mu,$ the mean additional time spent sleeping on weekend days for students at this college. The following table is similar to the table that appears in Example 12.4 . The "what you know" information has been provided. Complete the table by filling in the "how you know it" column.

Rashmi Sinha

Numerade Educator

02:08

Problem 9

Explain the difference between $\sigma$ and $\sigma_{\vec{x}}$.

Harsh Gadhiya

Numerade Educator

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Problem 10

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5.
a. What are the mean and standard deviation of the sampling distribution of $\bar{x}$ ? Describe the shape of the sampling distribution of $\bar{x}$.
b. What is the approximate probability that $\bar{x}$ will be within 0.5 of the population mean $\mu$ ?
c. What is the approximate probability that $\bar{x}$ will differ from $\mu$ by more than $0.7 ?$

Rashmi Sinha

Numerade Educator

04:39

Problem 11

A random sample is selected from a population with mean $\mu=200$ and standard deviation $\sigma=15$. Determine the mean and standard deviation of the sampling distribution of $\bar{x}$ for each of the following sample sizes:
a. $n=12$
d. $n=40$
b. $n=20$
e. $n=90$
c. $n=25$
f. $n=300$

Prabhakar Kumar

Numerade Educator

00:49

Problem 12

For which of the sample sizes given in the previous exercise would it be reasonable to think that the sampling distribution of $\bar{x}$ is approximately normal in shape?

Hossam Mohamed

Numerade Educator

00:40

Problem 13

Explain the difference between $\bar{x}$ and $\mu_{\vec{x}}$.

Katelyn Chen

Numerade Educator

03:00

Problem 14

A sign in the elevator of a college library indicates a limit of 16 persons. In addition, there is a weight limit of 2500 pounds. Assume that the average weight of students, faculty, and staff at this college is 150 pounds, that the standard deviation is 27 pounds, and that the distribution of weights of individuals on campus is approximately normal. A random sample of 16 persons from the campus will be selected.
a. What is the mean of the sampling distribution of $\bar{x} ?$
b. What is the standard deviation of the sampling distribution of $\bar{x} ?$
c. What average weights for a sample of 16 people will result in the total weight exceeding the weight limit of 2500 pounds?
d. What is the probability that a random sample of 16 people will exceed the weight limit?

Harsh Gadhiya

Numerade Educator

03:43

Problem 15

Suppose that the population mean value of interpupillary distance (the distance between the pupils of the left and right eyes) for adult males is $65 \mathrm{~mm}$ and that the population standard deviation is $5 \mathrm{~mm}$.
a. If the distribution of interpupillary distance is normal and a random sample of $n=25$ adult males is to be selected, what is the probability that the sample mean distance $\bar{x}$ for these 25 will be between 64 and $67 \mathrm{~mm}$ ? At least $68 \mathrm{~mm}$ ?
b. Suppose that a random sample of 100 adult males is to be selected. Without assuming that interpupillary distance is normally distributed, what is the approximate probability that the sample mean distance will be between 64 and 67 $\mathrm{mm}$ ? At least $68 \mathrm{~mm} ?$

Prabhakar Kumar

Numerade Educator

02:30

Problem 16

Suppose that a random sample of size 100 is to be drawn from a population with standard deviation 10 .
a. What is the probability that the sample mean will be within 20 of the value of $\mu$ ?
b. For this example $(n=100, \sigma=10)$, complete each of the following statements by calculating the appropriate value:
i. Approximately $95 \%$ of the time, $\bar{x}$ will be within of $\mu$.
ii. Approximately $0.3 \%$ of the time, $\bar{x}$ will be farther than from $\mu$.

Robin Corrigan

Numerade Educator

01:22

Problem 17

A manufacturing process is designed to produce bolts with a diameter of 0.5 inches. Once each day, a random sample of 36 bolts is selected and the bolt diameters are recorded. If the resulting sample mean is less than 0.49 inches or greater than 0.51 inches, the process is shut down for adjustment. The standard deviation of bolt diameters is 0.02 inches. What is the probability that the manufacturing line will be shut down unnecessarily? (Hint: Find the probability of observing an $\bar{x}$ in the shutdown range when the actual process mean is 0.5 inches.)

Linh Vu

Numerade Educator

07:15

Problem 18

An airplane with room for 100 passengers has a total baggage limit of 6000 pounds. Suppose that the weight of baggage checked by an individual passenger, $x$, has a mean of 50 pounds and a standard deviation of 20 pounds. If 100 passengers will board a flight, what is the approximate probability that the total weight of their baggage will exceed the limit? (Hint: With $n=100$, the total weight exceeds the limit when the mean weight $\bar{x}$ exceeds $6000 / 100 .$ )

Chris Trentman

Numerade Educator