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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. (Source: U.S. Dept. of Agriculture)

300

CHAPTER 7. THE CENTRAL LIMIT THEOREM

a. In words, X =

b. In words, X =

c. X~

d. The IQR for X is from _______ acres to _______ acres.

Exercise 7.7.17

(Solution on p. 315.)

The stock closing prices of 35 U.S. semiconductor manufacturers are given below. (Source: Wall

Street Journal)

8.625; 30.25; 27.625; 46.75; 32.875; 18.25; 5; 0.125; 2.9375; 6.875; 28.25; 24.25; 21; 1.5; 30.25; 71; 43.5;

49.25; 2.5625; 31; 16.5; 9.5; 18.5; 18; 9; 10.5; 16.625; 1.25; 18; 12.875; 7; 12.875; 2.875; 60.25; 29.25

a. In words, X =

b. i. x =

ii. sx =

iii. n =

c. Construct a histogram of the distribution of the averages. Start at x = −0.0005. Make bar

widths of 10.

d. In words, describe the distribution of stock prices.

e. Randomly average 5 stock prices together. (Use a random number generator.) Continue aver-

aging 5 pieces together until you have 10 averages. List those 10 averages.

f. Use the 10 averages from (e) to calculate:

i. x =

ii. sx =

g. Construct a histogram of the distribution of the averages. Start at x = −0.0005. Make bar

widths of 10.

h. Does this histogram look like the graph in (c)?

i. In 1 - 2 complete sentences, explain why the graphs either look the same or look different?

j. Based upon the theory of the Central Limit Theorem, X~

Exercise 7.7.18

Use the Initial Public Offering data (Section 13.3.2: Stock Prices) (see “Table of Contents) to do this

problem.

a. In words, X =

b. i. µ X =

ii. σ X =

iii. n =

c. Construct a histogram of the distribution. Start at x = −0.50. Make bar widths of $5.

d. In words, describe the distribution of stock prices.

e. Randomly average 5 stock prices together. (Use a random number generator.) Continue aver-

aging 5 pieces together until you have 15 averages. List those 15 averages.

f. Use the 15 averages from (e) to calculate the following:

i. x =

ii. sx =

g. Construct a histogram of the distribution of the averages. Start at x = −0.50. Make bar widths

of $5.

h. Does this histogram look like the graph in (c)? Explain any differences.

i. In 1 - 2 complete sentences, explain why the graphs either look the same or look different?

j. Based upon the theory of the Central Limit Theorem, X~

301

7.7.1 Try these multiple choice questions (Exercises19 - 23).

The next two questions refer to the following information: The time to wait for a particular rural bus

is distributed uniformly from 0 to 75 minutes. 100 riders are randomly sampled to learn how long they

waited.

Exercise 7.7.19

(Solution on p. 315.)

The 90th 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

Exercise 7.7.20

(Solution on p. 315.)

Would you be surprised, based upon numerical calculations, if the sample average wait time (in

minutes) for 100 riders was less than 30 minutes?

A. Yes

B. No

C. There is not enough information.

Exercise 7.7.21

(Solution on p. 315.)

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 one

C. The curve never touches the x-axis

D. The curve is skewed to the right

The next three questions refer to the following information: The cost of unleaded gasoline in the Bay Area

once followed an unknown distribution with a mean of $2.59 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.

Exercise 7.7.22

(Solution on p. 315.)

The distribution to use for the average cost of gasoline for the 16 gas stations is

A. X ∼ N (2.59, 0.10)

B. X ∼ N 2.59, 0.10

√16

C. X ∼ N 2.59, 0.10

D. X ∼ N 2.59, 16

0.10

Exercise 7.7.23

(Solution on p. 315.)

What is the probability that the average price for 16 gas stations is over $2.69?

A. Almost zero

B. 0.1587

C. 0.0943

D. Unknown

Exercise 7.7.24

(Solution on p. 315.)

Find the probability that the average price for 30 gas stations is less than $2.55.

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CHAPTER 7. THE CENTRAL LIMIT THEOREM

A. 0.6554

B. 0.3446

C. 0.0142

D. 0.9858

E. 0

Exercise 7.7.25

(Solution on p. 315.)

For the Charter School Problem (Example 3) in Central Limit Theorem: Using the Central Limit

Theorem, calculate the following using the normal approximation to the binomial.

A. Find the probability that less than 100 favor a charter school for grades K - 5.

B. Find the probability that 170 or more favor a charter school for grades K - 5.

C. Find the probability that no more than 140 favor a charter school for grades K - 5.

D. Find the probability that there are fewer than 130 that favor a charter school for grades K - 5.

E. Find the probability that exactly 150 favor a charter school for grades K - 5.

If you either have access to an appropriate calculator or computer software, try calculating these

probabilities using the technology. Try also using the suggestion that is at the bottom of Central

Limit Theorem: Using the Central Limit Theorem for finding a website that calculates binomial

probabilities.

Exercise 7.7.26

(Solution on p. 315.)

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.

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.

If you either have access to an appropriate calculator or computer software, try calculating these

probabilities using the technology. Try also using the suggestion that is at the bottom of Central

Limit Theorem: Using the Central Limit Theorem for finding a website that calculates binomial

probabilities.

**Exercise 24 contributed by Roberta Bloom

303

7.8 Review9

The next three questions refer to the following information: Richard’s Furniture Company delivers fur-

niture 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.

Exercise 7.8.1

(Solution on p. 316.)

X ∼

A. U (0, 4)

B. U (10, 2)

C. Exp (2)

D. N (2, 1)

Exercise 7.8.2

(Solution on p. 316.)

The average wait time is:

A. 1 hour

B. 2 hour

C. 2.5 hour

D. 4 hour

Exercise 7.8.3

(Solution on p. 316.)

Suppose that it is now past noon on a delivery day. The probability that a person must wait at

least 1 1 more hours is:

A. 14

B. 12

C. 34

D. 38

Exercise 7.8.4

(Solution on p. 316.)

Given: X~Exp 1 .

a. Find P (x > 1)

b. Calculate the minimum value for the upper quartile.

c. Find P x = 13

Exercise 7.8.5

(Solution on p. 316.)

• 40% of full-time students took 4 years to graduate

• 30% of full-time students took 5 years to graduate

• 20% of full-time students took 6 years to graduate

• 10% of full-time students took 7 years to graduate

The expected time for full-time students to graduate is:

A. 4 years

B. 4.5 years

C. 5 years

D. 5.5 years

9This content is available online at <http://cnx.org/content/m16955/1.12/>.

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CHAPTER 7. THE CENTRAL LIMIT THEOREM

Exercise 7.8.6

(Solution on p. 316.)

Which of the following distributions is described by the following example?

Many people can run a short distance of under 2 miles, but as the distance increases, fewer people

can run that far.

A. Binomial

B. Uniform

C. Exponential

D. Normal

Exercise 7.8.7

(Solution on p. 316.)

The length of time to brush one’s teeth is generally thought to be exponentially distributed with

a mean of 3 minutes. Find the probability that a randomly selected person brushes his/her teeth

less than 3 minutes.

A. 0.5

B. 34

C. 0.43

D. 0.63

Exercise 7.8.8

(Solution on p. 316.)

Which distribution accurately describes the following situation?

The chance that a teenage boy regularly gives his mother a kiss goodnight (and he should!!) is

about 20%. Fourteen teenage boys are randomly surveyed.

X =the number of teenage boys that regularly give their mother a kiss goodnight

A. B (14, 0.20)

B. P (2.8)

C. N (2.8, 2.24)

D. Exp

0.20

Exercise 7.8.9

(Solution on p. 316.)

Which distribution accurately describes the following situation?

A 2008 report on technology use states that approximately 20 percent of U.S. households have

never sent an e-mail.

(source: http://www.webguild.org/2008/05/20-percent-of-americans-

have-never-used-email.php) Suppose that we select a random sample of fourteen U.S. households

X =the number of households in a 2008 sample of 14 households that have never sent an email

A. B (14, 0.20)

B. P (2.8)

C. N (2.8, 2.24)

D. Exp

0.20

**Exercise 9 contributed by Roberta Bloom

305

7.9 Lab 1: Central Limit Theorem (Pocket Change)10

Class Time:

Names:

7.9.1 Student Learning Outcomes:

• The student will demonstrate and compare properties of the Central Limit Theorem.

NOTE: This lab works best when sampling from several classes and combining data.

7.9.2 Collect the Data

1. Count the change in your pocket. (Do not include bills.)

2. Randomly survey 30 classmates. Record the values of the change.

__________

Table 7.1

3. Construct a histogram. Make 5 - 6 intervals. Sketch the graph using a ruler and pencil. Scale the axes.

10This content is available online at <http://cnx.org/content/m16950/1.10/>.

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CHAPTER 7. THE CENTRAL LIMIT THEOREM

Figure 7.1

4. Calculate the following (n = 1; surveying one person at a time):

a. x =

b. s =

5. Draw a smooth curve through the tops of the bars of the histogram. Use 1 – 2 complete sentences to

describe the general shape of the curve.

7.9.3 Collecting Averages of Pairs

Repeat steps 1 - 5 (of the section above titled "Collect the Data") with one exception. Instead of recording

the change of 30 classmates, record the average change of 30 pairs.

1. Randomly survey 30 pairs of classmates. Record the values of the average of their change.

__________

Table 7.2

2. Construct a histogram. Scale the axes using the same scaling you did for the section titled "Collecting

the Data". Sketch the graph using a ruler and a pencil.

307

Figure 7.2

3. Calculate the following (n = 2; surveying two people at a time):

a. x =

b. s =

4. Draw a smooth curve through tops of the bars of the histogram. Use 1 – 2 complete sentences to

describe the general shape of the curve.

7.9.4 Collecting Averages of Groups of Five

Repeat steps 1 – 5 (of the section titled "Collect the Data") with one exception. Instead of recording the

change of 30 classmates, record the average change of 30 groups of 5.

1. Randomly survey 30 groups of 5 classmates.

Record the values of the average of their

change.

__________

Table 7.3

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CHAPTER 7. THE CENTRAL LIMIT THEOREM

2. Construct a histogram. Scale the axes using the same scaling you did for the section titled "Collect the

Data". Sketch the graph using a ruler and a pencil.

Figure 7.3

3. Calculate the following (n = 5; surveying five people at a time):

a. x =

b. s =

4. Draw a smooth curve through tops of the bars of the histogram. Use 1 – 2 complete sentences to

describe the general shape of the curve.

7.9.5 Discussion Questions

1. As n changed, why did the shape of the distribution of the data change? Use 1 – 2 complete sentences

to explain what happened.

2. In the section titled "Collect the Data", what was the approximate distribution of the data? X ∼

3. In the section titled "Collecting Averages of Groups of Five", what was the approximate distribution

of the averages? X ∼

4. In 1 – 2 complete sentences, explain any differences in your answers to the previous two questions.

309

7.10 Lab 2: Central Limit Theorem (Cookie Recipes)11

Class Time:

Names:

7.10.1 Student Learning Outcomes:

• The student will demonstrate and compare properties of the Central Limit Theorem.

7.10.2 Given:

X = length of time (in days) that a cookie recipe lasted at the Olmstead Homestead. (Assume that each of

the different recipes makes the same quantity of cookies.)

Recipe #

Table 7.4

Calculate the following:

a. µ x =

b. σ x =

11This content is available online at <http://cnx.org/content/m16945/1.11/>.

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CHAPTER 7. THE CENTRAL LIMIT THEOREM

7.10.3 Collect the Data

Use a random number generator to randomly select 4 samples of size n = 5 from the given population.

Record your samples below. Then, for each sample, calculate the mean to the nearest tenth. Record them in

the spaces provided. Record the sample means for the rest of the class.

1. Complete the table:

Sample 1

Sample 2

Sample 3

Sample 4

Sample means from other groups:

Means:

x =

Table 7.5

2. Calculate the following:

a. x =

b. sx =

3. Again, use a random number generator to randomly select 4 samples from the population. This time,

make the samples of size n = 10. Record the samples below. As before, for each sample, calculate the

mean to the nearest tenth. Record them in the spaces provided. Record the sample means for the rest

of the class.

Sample 1

Sample 2