Statistics

Barbara Illowsky, Susan Dean

Chapter 7

The Central Limit Theorem - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

Problem 1

Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let $X$ be the random variable representing the time
it takes her to complete one review. Assume $X$ is normally distributed. Let $X$ be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.What is the mean, standard deviation, and sample size?

Bryan Meares

Bryan Meares

Numerade Educator

Problem 2

Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let $X$ be the random variable representing the time
it takes her to complete one review. Assume $X$ is normally distributed. Let $X$ be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.Complete the distributions.
a. $X \sim$ ______(______,_______).
b. $\quad \bar{x} \sim$ _______( _____,______)

Tony Wilson

Numerade Educator

Problem 3

Find the probability that one review will take Yoonie from 3.5 to 4.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.b. $P(_______$ $<x<_________)=__________.

Sheryl Ezze

Numerade Educator

Problem 4

Find the probability that the mean of a month's reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.b. $P(_________________)=__________.

Karen Song

Numerade Educator

Problem 5

What causes the probabilities in Exercise 7.3 and Exercise 7.4 to be different?

Karen Song

Numerade Educator

Problem 6

Find the $95^{\text {th }}$ percentile for the mean time to complete one month's reviews. Sketch the graph.b. The $95^{\text {th }}$ percentile $=$ ____________ .

Karen Song

Numerade Educator

Problem 7

Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of $12 .$ A sample size of 95 is drawn randomly from the population.
Find the probability that the sum of the 95 values is greater than 7,650 .

Bryan Meares

Numerade Educator

Problem 8

Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of $12 .$ A sample size of 95 is drawn randomly from the population.Find the probability that the sum of the 95 values is less than 7,400 .

Bryan Meares

Numerade Educator

Problem 9

Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of $12 .$ A sample size of 95 is drawn randomly from the population.Find the sum that is two standard deviations above the mean of the sums.

Bryan Meares

Numerade Educator

Problem 10

Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of $12 .$ A sample size of 95 is drawn randomly from the population.
Find the sum that is 1.5 standard deviations below the mean of the sums.

Robin Corrigan

Numerade Educator

Problem 11

Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has mean of 180 and a standard deviation of $20 .$ A sample size of 40 is drawn randomly. Find the probability that the sum of the 40 values is greater than 7,500 .

James Kiss

Numerade Educator

Problem 12

Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has mean of 180 and a standard deviation of $20 .$ A sample size of 40 is drawn randomly.Find the probability that the sum of the 40 values is less than 7,000 .

James Kiss

Numerade Educator

Problem 13

Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has mean of 180 and a standard deviation of $20 .$ A sample size of 40 is drawn randomly.Find the sum that is one standard deviation above the mean of the sums.

James Kiss

Numerade Educator

Problem 14

Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has mean of 180 and a standard deviation of $20 .$ A sample size of 40 is drawn randomly.Find the sum that is one standard deviation above the mean of the sums.

James Kiss

Numerade Educator

Problem 15

Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has mean of 180 and a standard deviation of $20 .$ A sample size of 40 is drawn randomly.Find the percentage of sums between 1.5 standard deviations below the mean of the sums and one standard deviation above the mean of the sums.

James Kiss

Numerade Educator

Problem 16

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of $0.5 .$ The researcher randomly selects a sample of 100.Find the probability that the sum of the 100 values is greater than 3,910 .

James Kiss

Numerade Educator

Problem 17

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of $0.5 .$ The researcher randomly selects a sample of 100.Find the probability that the sum of the 100 values is less than 3,900 .

James Kiss

Numerade Educator

Problem 18

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of $0.5 .$ The researcher randomly selects a sample of 100.Find the probability that the sum of the 100 values falls between the numbers you found in (16) and (17).

James Kiss

Numerade Educator

Problem 19

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of $0.5 .$ The researcher randomly selects a sample of 100.Find the sum with a z-score of $-2.5 .$.

James Kiss

Numerade Educator

Problem 20

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of $0.5 .$ The researcher randomly selects a sample of 100.Find the sum with a z-score of 0.5.

James Kiss

Numerade Educator

Problem 21

Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same type of soda. The mean is 39.01 with a standard deviation of $0.5 .$ The researcher randomly selects a sample of 100.Find the probability that the sums will fall between the $z$ -scores -2 and 1 .

James Kiss

Numerade Educator

Problem 22

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.What is the mean of $\Sigma X$ ?

James Kiss

Numerade Educator

Problem 23

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.What is the standard deviation of $\Sigma X$ ?

James Kiss

Numerade Educator

Problem 24

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.What is $P(\Sigma x=290) ?$

James Kiss

Numerade Educator

Problem 25

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.What is $P(\Sigma x>290) ?$

James Kiss

Numerade Educator

Problem 26

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.26. True or False: Only the sums of normal distributions are also normal distributions.In order for the sums of a distribution to approach a normal distribution, what must be true?

Ryan Mcalister

Numerade Educator

Problem 27

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.In order for the sums of a distribution to approach a normal distribution, what must be true?

James Kiss

Numerade Educator

Problem 28

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.What three things must you know about a distribution to find the probability of sums?

James Kiss

Numerade Educator

Problem 29

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.An unknown distribution has a mean of 25 and a standard deviation of six. Let $X=$ one object from this distributio What is the sample size if the standard deviation of $\Sigma X$ is $42 ?$

Karen Song

Numerade Educator

Problem 30

Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let $X=$ the object of interest.An unknown distribution has a mean of 19 and a standard deviation of $20 .$ Let $X=$ the object of interest. What is the sample size if the mean of $\Sigma X$ is $15,200 ?$

James Kiss

Numerade Educator

Problem 31

Use the following information to answer the next three exercises: A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of $0.7 .$ She samples 400 customers.What is the z-score for $\Sigma x=840$ ?

James Kiss

Numerade Educator

Problem 32

Use the following information to answer the next three exercises: A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of $0.7 .$ She samples 400 customers.What is the $z$ -score for $\Sigma x=1,186 ?$

James Kiss

Numerade Educator

Problem 33

Use the following information to answer the next three exercises: A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of $0.7 .$ She samples 400 customers.What is $P(\Sigma x<1186) ?$

James Kiss

Numerade Educator

Problem 34

Use the following information to answer the next three exercises: An unkwon distribution has a mean of $100,$ a standard deviation of $100,$ and a sample size of $100 .$ Let $X=$ one object of interest.
What is the mean of $\Sigma X$ ?

James Kiss

Numerade Educator

Problem 35

Use the following information to answer the next three exercises: An unkwon distribution has a mean of $100,$ a standard deviation of $100,$ and a sample size of $100 .$ Let $X=$ one object of interest.What is the standard deviation of $\Sigma X$ ?

James Kiss

Numerade Educator

Problem 36

Use the following information to answer the next three exercises: An unkwon distribution has a mean of $100,$ a standard deviation of $100,$ and a sample size of $100 .$ Let $X=$ one object of interest.What is $P(\Sigma x>9000) ?$

James Kiss

Numerade Educator

Problem 37

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.a. What is the distribution for the weights of one 25 -pound lifting weight? What are the mean and standard deivation?
b. What is the distribution for the mean weight of 100 25-pound lifting weights?
c. Find the probability that the mean actual weight for the 100 weights is less than $24.9 .$

Sheryl Ezze

Numerade Educator

Problem 38

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Draw the graph of Exercise 7.37.

Sheryl Ezze

Numerade Educator

Problem 39

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Find the probability that the mean actual weight for the 100 weights is greater than 25.2 .

Willis James

Numerade Educator

Problem 40

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Draw the graph of Exercise 7.39.

Sheryl Ezze

Numerade Educator

Problem 41

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Find the 90 $^{\text {th }}$ percentile for the mean weight for the 100 weights.

Sherrie Fenner

Numerade Educator

Problem 42

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Draw the graph of Exercise 7.41.

Sherrie Fenner

Numerade Educator

Problem 43

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.a. What is the distribution for the sum of the weights of 100 25-pound lifting weights?b. Find $P(\Sigma x<2450)$

Sherrie Fenner

Numerade Educator

Problem 44

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Draw the graph of Exercise 7.43.

Sherrie Fenner

Numerade Educator

Problem 45

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Find the $90^{\text {th }}$ percentile for the total weight of the 100 weights.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 46

A manufacturer produces 25 -pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely, so the distribution of weights is uniform. A sample of 100 weights is taken.Draw the graph of Exercise 7.45.

Sherrie Fenner

Numerade Educator

Problem 47

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.a. What is the standard deviation?
b. What is the parameter $m$ ?

Sherrie Fenner

Numerade Educator

Problem 48

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.What is the distribution for the length of time one battery lasts?

Sherrie Fenner

Numerade Educator

Problem 49

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.What is the distribution for the mean length of time 64 batteries last?

Sherrie Fenner

Numerade Educator

Problem 50

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.What is the distribution for the total length of time 64 batteries last?

Sherrie Fenner

Numerade Educator

Problem 51

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken. Find the probability that the sample mean is between 7 and 11 .

Sherrie Fenner

Numerade Educator

Problem 52

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken. Find the $80^{\text {th }}$ percentile for the total length of time 64 batteries last.

Sherrie Fenner

Numerade Educator

Problem 53

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken. Find the interquartile range (IQR) for the mean amount of time 64 batteries last.

Sherrie Fenner

Numerade Educator

Problem 54

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken. Find the middle 80 percent for the total amount of time 64 batteries last.

Sherrie Fenner

Numerade Educator

Problem 55

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.Find $P(\Sigma x>420)$

Sherrie Fenner

Numerade Educator

Problem 56

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. Find the $90^{\text {th }}$ percentile for the sums.

Sherrie Fenner

Numerade Educator

Problem 57

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. Find the $15^{\text {th }}$ percentile for the sums.

Sherrie Fenner

Numerade Educator

Problem 58

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. Find the first quartile for the sums.

AG

Ankit Gupta

Numerade Educator

Problem 59

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. Find the third quartile for the sums.

Tony Wilson

Numerade Educator

Problem 60

Use the following information to answer the next six exercises: A uniform distribution has a minimum of six and a maximum of ten. Find the $80^{\text {th }}$ percentile for the sums.

Sherrie Fenner

Numerade Educator

Problem 61

Previously, De Anza's statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $\$ 0.88 .$ Suppose that we randomly pick 25 daytime statistics students.
a. In words, $X=$ ____________ .
b. $X \sim$ ________ (_______,_______)
c. In words, $X$ =_________ .
d. $\quad \bar{X} \sim$ = _______ (______ ,______ ).
e. Find the probability that an individual had between $\$ 0.80$ and $\$ 1.00 .$ Graph the situation, and shade in the area to be determined.
f. Find the probability that the average amount of change of the 25 students was between $\$ 0.80$ and $\$ 1.00 .$ Graph the situation, and shade in the area to be determined.
g. Explain why there is a difference in part (e) and part (f).

Karen Song

Numerade Educator

Problem 62

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
a. If $X=$ average distance in feet for 49 fly balls, then $X \sim$_________( _____ , _____ ).
b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for $X$. Shade the region corresponding to the probability. Find the probability.
c. Find the $80^{\text {th }}$ percentile of the distribution of the average of 49 fly balls.

Nick Johnson

Numerade Educator

Problem 63

According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample 36 taxpayers.
a. In words, $X=$ _________ .
b. In words, $X=$ _________ .
c. $X \sim$ ________(______ , ____ ).
d. Would you be surprised if the 36 taxpayers finished their Form 1040 s in an average of more than 12 hours? Explain why or why not in complete sentences.
e. Would you be surprised if one taxpayer finished his or her Form 1040 in more than 12 hours? In a complete sentence, explain why.

Karen Song

Numerade Educator

Problem 64

Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Let $X$ be the average of the 49 races.
a. $\quad \bar{X} \sim$______ ( _____,_______) .
b. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.
c. Find the $80^{\text {th }}$ percentile for the average of these 49 marathons.
d. Find the median of the average running times.

Karen Song

Numerade Educator

Problem 65

The length of songs in a collector's online album collection is uniformly distributed from 2 to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.
a. In words, $X=$ _______ .
b. $X \sim$ ________ .
c. In words, $X=$ _______ .
d. $\quad \bar{X} \sim$ _______ ( _____,_____).
e. Find the first quartile for the average song length.
f. The IQR for the average song length is _______ - _______ .

Karen Song

Numerade Educator

Problem 66

In $1940,$ the average size of a U.S. farm was 174 acres. Let's say that the standard deviation was 55 acres. Suppose we randomly survey 38 farmers from 1940.
a. In words, $X=$ _________ .
b. In words, $X=$ _______ .
c. $X \sim$ __________ ( _____ , ______ ) .
d. The IQR for $X$ is from _______ acres to _____ acres.

Karen Song

Numerade Educator

Problem 67

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.
a. When the sample size is large, the mean of $X$ is approximately equal to the mean of $X$.
b. When the sample size is large, $X$ is approximately normally distributed.
c. When the sample size is large, the standard deviation of $X$ is approximately the same as the standard deviation of $X$.

Karen Song

Numerade Educator

Problem 68

The percentage of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about ten. Suppose that 16 individuals are randomly chosen. Let $X=$ average percentage of fat calories.
a. $\quad \bar{X} \sim$ _________ ( ______ , _____ )
b. For the group of $16,$ find the probability that the average percentage of fat calories consumed is more than five. Graph the situation and shade in the area to be determined.
c. Find the first quartile for the average percentage of fat calories. .

Karen Song

Numerade Educator

Problem 69

The distribution of income in some economically developing countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge-shaped distribution. Let the average salary be $\$ 2,000$ per year with a standard deviation of $\$ 8,000$. We randomly survey 1,000 residents of that country.
a. In words, $X=$ ______ .
b. In words, $X=$ ________ .
c. $\quad X \sim$ ________ ( ______ , ______ ) .
d. How is it possible for the standard deviation to be greater than the average?
e. Why is it more likely that the average salary of the 1,000 residents will be from $\$ 2,000$ to $\$ 2,100$ than from $\$ 2,1 C$ to $\$ 2,200 ?$

Karen Song

Numerade Educator

Problem 70

Which of the following is NOT true about the distribution for averages?
a. The mean, median, and mode are equal.
b. The area under the curve is 1.
c. The curve never touches the $x$ -axis.
d. The curve is skewed to the right.

James Kiss

Numerade Educator

Problem 71

The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $\$ 4.59$ and standard deviation of $\$ 0.10 .$ Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. The distribution to use for the average cost of gasoline for the 16 gas stations is:
a. $\quad \bar{X} \sim N(4.59,0.10)$
b. $\quad \bar{X} \sim N\left(4.59, \frac{0.10}{\sqrt{16}}\right)$
c. $\quad \bar{X} \sim N\left(4.59, \frac{16}{0.10}\right)$
d. $\quad \bar{X} \sim N\left(4.59, \frac{\sqrt{16}}{0.10}\right)$

Karen Song

Numerade Educator

Problem 72

Which of the following is NOT true about the theoretical distribution of sums?
a. The mean, median, and mode are equal.
b. The area under the curve is one.
c. The curve never touches the $x$ -axis.
d. The curve is skewed to the right.

James Kiss

Numerade Educator

Problem 73

Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of seven days. We randomly sample nine trials.
a. In words, $\Sigma X=$ ________ .
b. $\Sigma X \sim$ ______( _____ , _____ ) .
c. Find the probability that the total length of the nine trials is at least 225 days.
d. Ninety percent of the total of nine of these types of trials will last at least how long?

Bryan Meares

Numerade Educator

Problem 74

Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and standard deviation of $1.1547 .$ We randomly survey 64 homes with children.
a. In words, $X=$ _______ .
b. The distribution is _______ .
c. In words, $\Sigma X=$ _____ .
d. $\Sigma X \sim$ _______ ( ____ , ____ ) .
e. Find the probability that the total weight of the open boxes is less than 250 pounds.
f. Find the $35^{\text {th }}$ percentile for the total weight of open boxes of cereal.

Ahmad Reda

Numerade Educator

Problem 75

Salaries for entry-level managers at a restaurant chain are normally distributed with a mean of $\$ 44,000$ and a standard deviation of $\$ 6,500 .$ We randomly survey 10 managers from these restaurants.
a. In words, $X=$ _______ .
b. $X \sim$ _______ ( ____ , ____ ) .
c. In words, $\Sigma X=$ _______ .
d. $\Sigma X \sim$ _______ ( ____ , ____ ) .
e. Find the probability that the managers earn a total of over $\$ 400,000$.
f. Find the 90" percentile for an individual manager's salary.
g. Find the $90^{\text {th }}$ percentile for the sum of ten managers' salary.
h. If we surveyed 70 managers instead of ten, graphically, how would that change the distribution in part (d)?
i. If each of the 70 managers received a $\$ 3,000$ raise, graphically, how would that change the distribution in par
(b)?

Sheryl Ezze

Numerade Educator

Problem 76

The attention span of a two-year-old is exponentially distributed with a mean of about eight minutes. Suppose we randomly survey 60 two-year-olds.
a. In words, $X=$ _______ .
b. $X \sim$ _______ ( ____ , ____ ) .
c. In words, $X=$ _______ .
d. $\quad \bar{X} \sim$ _______ ( ____ , ____ ) .
e. Before doing any calculations, which do you think will be higher? Explain why.
i. The probability that an individual attention span is less than 10 minutes.
ii. The probability that the average attention span for the 60 children is less than 10 minutes.
f. Calculate the probabilities in part (e).
g. Explain why the distribution for $X$ is not exponential.

Sherrie Fenner

Numerade Educator

Problem 77

The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows: a. In words, $X=$ _______ .
b. i. $\quad x=$ ______ .
ii. $\quad s_{x}=$ _______ .
iii. $\quad n=$ ________ .
c. Construct a histogram of the distribution of the averages. Start at $x=-0.0005 .$ Use bar widths of $10 .$
d. In words, describe the distribution of the stock prices.
e. Randomly average five stock prices together. (Use a random number generator.) Continue averaging five prices together until you have 10 averages. List those 10 averages.
f. Use the 10 averages from part (e) to calculate the following:
i. $\quad x=$ ________ .
ii. $\quad S_{X}=$ ______ .
g. Construct a histogram of the distribution of the averages. Start at $x=-0.0005 .$ Use bar widths of $10 .$
h. Does this histogram look like the graph in Part (c)?
i. In one or two complete sentences, explain why the graphs either look the same or look different.
j. Based on the theory of the central limit theorem, $X \sim$ _______ ( ____ ,______ ).

Jerelyn Nevil

Numerade Educator

Problem 78

Use the following information to answer the next three exercises: Richard's Furniture Company delivers furniture from 10 a.m. to 2 p.m. continuously and uniformly. We are interested in how long (in hours) past the 10 a.m. start time that individuals wait for their delivery.
$X \sim$_______ ( ____ , ____ ) .
a. $\quad U(0,4)$
b. $U(10,2)$
c. $\quad E \chi p(2)$
d. $\quad N(2,1)$

Sherrie Fenner

Numerade Educator

Problem 79

The average wait time is:
a. one hour
b. two hours
c. two and a half hours
d. four hours

Hubert Agamasu

Numerade Educator

Problem 80

Suppose that it is now past noon on a delivery day. The probability that a person must wait at least one and a half more hours is
a. $\quad \frac{1}{4}$
b. $\frac{1}{2}$
c. $\frac{3}{4}$
d. $\frac{3}{8}$

Sherrie Fenner

Numerade Educator

Problem 81

Use the following information to answer the next two exercises: The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited.The $90^{\text {th }}$ percentile sample average wait time (in minutes) for a sample of 100 riders is:
a. 315.0
b. 40.3
c. 38.5
d. 65.2

Sherrie Fenner

Numerade Educator

Problem 82

Use the following information to answer the next two exercises: The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited.82. Would you be surprised, based on numerical calculations, if the sample average wait time (in minutes) for 100 riders was less than 30 minutes? a. yes
b. no There is not enough information.

Sherrie Fenner

Numerade Educator

Problem 83

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $\$ 4.59$ and a standard deviation of $\$ 0.10 .$ Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.What's the approximate probability that the average price for 16 gas stations is more than $\$ 4.69 ?$
a. almost zero
b. 0.1587
c. 0.0943
d. unknown

Sherrie Fenner

Numerade Educator

Problem 84

Use the following to answer the next two exercises: The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $\$ 4.59$ and a standard deviation of $\$ 0.10 .$ Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.Find the probability that the average price for 30 gas stations is less than $\$ 4.55$
a. 0.6554
b. 0.3446
c. 0.0142
d. 0.9858

Sherrie Fenner

Numerade Educator

Problem 85

Suppose in a local kindergarten through $12^{\text {th }}$ grade (K-12) school district, 53 percent of the population favor a charter school for grades K through five. A simple random sample of 300 is surveyed. Calculate the following using the normal approximation to the binomial distribtion.
a. Find the probability that less than 100 favor a charter school for grades K through 5 .
b. Find the probability that 170 or more favor a charter school for grades $K$ through 5 .
c. Find the probability that no more than 140 favor a charter school for grades $K$ through 5 .
d. Find the probability that there are fewer than 130 that favor a charter school for grades $K$ through 5 .
e. Find the probability that exactly 150 favor a charter school for grades $K$ through 5 .

Sherrie Fenner

Numerade Educator

Problem 86

Four friends, Janice, Barbara, Kathy, and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for 96 days. Use the normal approximation to the binomial to calculate the following probabilities. Round the standard deviation to four decimal places.
a. Find the probability that Janice is the driver at most 20 days.
b. Find the probability that Roberta is the driver more than 16 days.
c. Find the probability that Barbara drives exactly 24 of those 96 days.

Sherrie Fenner

Numerade Educator

Problem 87

$X \sim N(60,9) .$ Suppose that you form random samples of 25 from this distribution. Let $X$ be the random variable of averages. Let $\Sigma X$ be the random variable of sums. For parts (c) through (f), sketch the graph, shade the region, label and
scale the horizontal axis for $X$, and find the probability.
a. Sketch the distributions of $X$ and $X$ on the same graph.
b. $\quad \bar{X} \sim$ _______ ( ____ , ____ ) .
c. $P(x<60)=$ ________ .
d. Find the $30^{\text {th }}$ percentile for the mean.
e. $P(56<x<62)=$ ________ .
f. $P(18<x<58)=$ ________ .
g. $\quad \Sigma x \sim$ _______ ( ____ , ____ ) .
h. Find the minimum value for the upper quartile for the sum.
i. $\quad P(1400<\Sigma x<1550)=$ _______.

Sheryl Ezze

Numerade Educator

Problem 88

Suppose that the length of research papers is uniformly distributed from 10 to 25 pages. We survey a class in which 55 research papers were turned in to a professor. The 55 research papers are considered a random collection of all papers. We are interested in the average length of the research papers.
a. In words, $X=$ __________ .
b. $X \sim$ _______ .
c. $\mu_{x}=$ ______ .
d. $\quad \sigma_{x}=$ ______ .
e. In words, $X=$ ______.
f. $\quad \bar{X} \sim$_______ ( ____ , ____ ) .
g. In words, $\Sigma X=$ ________ .
h. $\sum X \sim$ _______ ( ____ , ____ ) .
i. Without doing any calculations, do you think that it's likely the professor will need to read a total of more than 1,050 pages? Why?
j. Calculate the probability that the professor will need to read a total of more than 1,050 pages.
k. Why is it so unlikely that the average length of the papers will be less than 12 pages?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 89

Salaries for managers in a restaurant chain are normally distributed with a mean of $\$ 44,000$ and a standard deviation of \$6,500. We randomly survey 10 managers from that district.
a. Find the $90^{\text {th }}$ percentile for an individual manager's salary.
b. Find the $90^{\text {th }}$ percentile for the average manager's salary.

Sheryl Ezze

Numerade Educator

Problem 90

The average length of a maternity stay in a U.S. hospital is said to be 2.4 days with a standard deviation of 0.9 days. We randomly survey 80 women who recently bore children in a U.S. hospital.
a. In words, $X=$ ________ .
b. In words, $X=$ _______ .
c. $\quad X \sim$ ________ .
d. In words, $\Sigma X=$ _______ .
e. $\Sigma X \sim$ _______ ( ____ , ____ ) .
f. Is it likely that an individual stayed more than five days in the hospital? Why or why not?
g. Is it likely that the average stay for the 80 women was more than five days? Why or why not?
h. Which is more likely:
i. An individual stayed more than five days.
ii. The average stay of 80 women was more than five days.
i. If we were to sum up the women's stays, is it likely that collectively, they spent more than a year in the hospital? Why or why not?

Sherrie Fenner

Numerade Educator

Problem 91

Provide graphs and use a calculator. Ready batteries has engineered a newer, longer-lasting AAA battery. The company claims this battery has an average life span of 17 hours with a standard deviation of 0.8 hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span is 16.7 hours. If the process is working properly, what is the probability of getting a random sample of 30 batteries in which the sample mean life span is 16.7 hours or less? Is the company's claim reasonable?

Sherrie Fenner

Numerade Educator

Problem 92

Provide graphs and use a calculator.Men have an average weight of 172 pounds with a standard deviation of 29 pounds.a. Find the probability that 20 randomly selected men will have a sum weight greater than 3,600 pounds.
b. If 20 men have a sum weight greater than 3,500 pounds, then their total weight exceeds the safety limits for water taxis. Based on (a), is this a safety concern? Explain.

Sherrie Fenner

Numerade Educator

Problem 93

Large bags of a brand of multicolored candies have a claimed net weight of 396.9 g. The standard deviation for the weight of the individual candies is 0.017 g. The following table is from a stats experiment conducted by a statistics class.The bag contained 465 candies and the listed weights in the table came from randomly selected candies. Count the weights.
a. Find the mean sample weight and the standard deviation of the sample weights of candies in the table.
b. Find the sum of the sample weights in the table and the standard deviation of the sum of the weights.
c. If 465 candies are randomly selected, find the probability that their weights sum to at least $396.9 \mathrm{g}$.
d. Is the candy company's labeling accurate?

Carly Stoner

Numerade Educator

Problem 94

The Screw Right Company claims their $\frac{3}{4}$ inch screws are within ±0.23 of the claimed mean diameter of 0.750 inches
with a standard deviation of 0.115 inches. The following data were recorded.The screws were randomly selected from the local home repair store.
a. Find the mean diameter and standard deviation for the sample.
b. Find the probability that 50 randomly selected screws will be within the stated tolerance levels. Is the company's diameter claim plausible?

Sherrie Fenner

Numerade Educator

Problem 95

Provide graphs and use a calculator.Your company has a contract to perform preventive maintenance on thousands of air conditioners in a large city. Based on service records from previous years, the time that a technician spends servicing a unit averages one hour with a standard deviation of one hour. In the coming week, your company will service a simple random sample of 70 units in the city. You plan to budget an average of 1.1 hours per technician to complete the work. Will this be enough time?

Sherrie Fenner

Numerade Educator

Problem 96

A typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ test, what is the probability that the sample mean scores will be between 85 and 125 points?

Sherrie Fenner

Numerade Educator

Problem 97

Certain coins have an average weight of 5.201 g with a standard deviation of 0.065 g. If a vending machine is designed to accept coins whose weights range from 5.111 g to 5.291 g, what is the expected number of rejected coins when 280 randomly selected coins are inserted into the machine?

Sherrie Fenner

Numerade Educator