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Introductory Statistics

Barbara Illowsky, Susan Dean

Chapter 11

The Chi-Square Distribution

Educators


Problem 1

If the number of degrees of freedom for a chi-square distribution is 25, what is the population mean and standard deviation?

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Problem 2

If $d f>90,$ the distribution is _______ . If $d f=15,$ the distribution is ________.

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Problem 3

When does the chi-square curve approximate a normal distribution?

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Problem 4

Where is $\mu$ located on a chi-square curve?

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Problem 5

Is it more likely the $d f$ is $90,20$ , or two in the graph?

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Problem 6

Determine the appropriate test to be used.
An archeologist is calculating the distribution of the frequency of the number of artifacts she finds in a dig site. Based on previous digs, the archeologist creates an expected distribution broken down by grid sections in the dig site. Once the site has been fully excavated, she compares the actual number of artifacts found in each grid section to see if her expectation was accurate.

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Problem 7

Determine the appropriate test to be used.
An economist is deriving a model to predict outcomes on the stock market. He creates a list of expected points on the stock market index for the next two weeks. At the close of each day’s trading, he records the actual points on the index. He wants to see how well his model matched what actually happened.

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Problem 8

Determine the appropriate test to be used.
A personal trainer is putting together a weight-lifting program for her clients. For a 90-day program, she expects each client to lift a specific maximum weight each week. As she goes along, she records the actual maximum weights her clients lifted. She wants to know how well her expectations met with what was observed.

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Problem 9

A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table 11.27.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Proportion }} \\ \hline A & {0.25} \\ \hline B & {0.30} \\ \hline C & {0.35} \\ \hline D & {0.10} \\ \hline\end{array}$$
The actual distribution for a class of 20 is in Table 11.28.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Frequency }} \\ \hline A & {7} \\ \hline B & {7} \\ \hline B & {7} \\ \hline C & {5} \\ \hline D & {1} \\ \hline\end{array}$$

$d f=$_____

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Problem 10

A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table 11.27.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Proportion }} \\ \hline A & {0.25} \\ \hline B & {0.30} \\ \hline C & {0.35} \\ \hline D & {0.10} \\ \hline\end{array}$$
The actual distribution for a class of 20 is in Table 11.28.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Frequency }} \\ \hline A & {7} \\ \hline B & {7} \\ \hline B & {7} \\ \hline C & {5} \\ \hline D & {1} \\ \hline\end{array}$$

State the null and alternative hypotheses.

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Problem 11

A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table 11.27.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Proportion }} \\ \hline A & {0.25} \\ \hline B & {0.30} \\ \hline C & {0.35} \\ \hline D & {0.10} \\ \hline\end{array}$$
The actual distribution for a class of 20 is in Table 11.28.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Frequency }} \\ \hline A & {7} \\ \hline B & {7} \\ \hline B & {7} \\ \hline C & {5} \\ \hline D & {1} \\ \hline\end{array}$$

$\chi^{2}$ test statistic $=$________

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Problem 12

A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table 11.27.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Proportion }} \\ \hline A & {0.25} \\ \hline B & {0.30} \\ \hline C & {0.35} \\ \hline D & {0.10} \\ \hline\end{array}$$
The actual distribution for a class of 20 is in Table 11.28.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Frequency }} \\ \hline A & {7} \\ \hline B & {7} \\ \hline B & {7} \\ \hline C & {5} \\ \hline D & {1} \\ \hline\end{array}$$

$p$ -value $=$ ______

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Problem 13

A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table 11.27.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Proportion }} \\ \hline A & {0.25} \\ \hline B & {0.30} \\ \hline C & {0.35} \\ \hline D & {0.10} \\ \hline\end{array}$$
The actual distribution for a class of 20 is in Table 11.28.
$$\begin{array}{|l|l|}\hline \text { Grade } & {\text { Frequency }} \\ \hline A & {7} \\ \hline B & {7} \\ \hline B & {7} \\ \hline C & {5} \\ \hline D & {1} \\ \hline\end{array}$$

At the 5% significance level, what can you conclude?

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Problem 14

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

If the ethnicities of AIDS victims followed the ethnicities of the total county population, fill in the expected number of cases per ethnic group. Perform a goodness-of-fit test to determine whether the occurrence of AIDS cases follows the ethnicities of the general population of Santa Clara County.

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Problem 15

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

$H_{0} :$ _____

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Problem 16

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

$H_{a} :$ ______

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Problem 17

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

Is this a right-tailed, left-tailed, or two-tailed test?

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Problem 18

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

degrees of freedom $=$ _____

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Problem 19

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

$\chi^{2}$ test statistic $=$ ______

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Problem 20

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

$p$ -value $=$ ______

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Problem 21

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the p-value.
Let $\alpha=0.05$
Decision: ________
Reason for the Decision: ___________
Conclusion (write out in complete sentences): __________

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Problem 22

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as in Table 11.29.
$$\begin{array}{|l|l|}\hline \text { Ethnicity } & {\text { Number of Cases }} \\ \hline \text { White } & {2,229} \\ \hline \text { Hispanic } & {1,157} \\ \hline \text { Black/African-American } & {457} \\ \hline \text { Asian, Pacific Islander } & {232} \\ \hline \text { Total } & {=4,075} \\ \hline\end{array}$$
The percentage of each ethnic group in Santa Clara County is as in Table 11.30.

Does it appear that the pattern of AIDS cases in Santa Clara County corresponds to the distribution of ethnic groups in this county? Why or why not?

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Problem 23

A pharmaceutical company is interested in the relationship between age and presentation of symptoms for a common viral infection. A random sample is taken of 500 people with the infection across different age groups.

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Problem 24

Determine the appropriate test to be used.
The owner of a baseball team is interested in the relationship between player salaries and team winning percentage. He takes a random sample of 100 players from different organizations.

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Problem 25

Determine the appropriate test to be used.
A marathon runner is interested in the relationship between the brand of shoes runners wear and their run times. She takes a random sample of 50 runners and records their run times as well as the brand of shoes they were wearing.

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Problem 26

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

State the hypotheses.
$H_{0} :$ ______
$H_{a}$ _______

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Problem 27

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

$d f=$ _____

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Problem 28

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

How many passengers are expected to travel between 201 and 300 miles and purchase second-class tickets?

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Problem 29

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

How many passengers are expected to travel between 401 and 500 miles and purchase first-class tickets?

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Problem 30

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

What is the test statistic?

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Problem 31

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

What is the $p$ -value?

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Problem 32

Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

What can you conclude at the 5% level of significance?

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Problem 33

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

Complete the table. African American, Native Hawaiian, Latino, Japanese Americans, White,
Smoking Level
Per Day
$1-10$
$11-20$
$21-20$
$31+$
TOTALS

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Problem 34

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

State the hypotheses.
$H_{0} :$ _____
$H_{a}$ ______

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Problem 35

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

Enter expected values in Table 11.32. Round to two decimal places. Calculate the following values:

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Problem 36

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

$d f=$ ______

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Problem 37

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

$\chi^{2}$ test statistic $=$ ________

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Problem 38

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

p-value $=$ ________

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Problem 39

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

Is this a right-tailed, left-tailed, or two-tailed test? Explain why.

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Problem 40

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the p-value.

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Problem 41

State the decision and conclusion (in a complete sentence) for the following preconceived levels of $\alpha .$

$\alpha=0.05$
a. Decision: _________
b. Reason for the decision:___________
c. Conclusion (write out in a complete sentence): ____________

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Problem 42

State the decision and conclusion (in a complete sentence) for the following preconceived levels of $\alpha .$

$\alpha=0.01$
a. Decision: ________
b. Reason for the decision:_______
C. Conclusion (write out in a complete sentence):________

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Problem 43

A math teacher wants to see if two of her classes have the same distribution of test scores. What test should she use?

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Problem 44

What are the null and alternative hypotheses for Exercise 11.43?

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Problem 45

A market researcher wants to see if two different stores have the same distribution of sales throughout the year. What type of test should he use?

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Problem 46

A meteorologist wants to know if East and West Australia have the same distribution of storms. What type of test should she use?

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Problem 47

What condition must be met to use the test for homogeneity?

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Problem 48

Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table 11.33.
$$\begin{array}{|l|l|l|l|l|l|}\hline & {20-30} & {30-40} & {40-50} & {50-60} \\ \hline \text { Private Practice } & {16} & {40} & {38} & {6} \\ \hline \text { Hospital } & {8} & {44} & {59} & {39} \\ \hline\end{array}$$

State the null and alternative hypotheses.

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Problem 49

Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table 11.33.
$$\begin{array}{|l|l|l|l|l|l|}\hline & {20-30} & {30-40} & {40-50} & {50-60} \\ \hline \text { Private Practice } & {16} & {40} & {38} & {6} \\ \hline \text { Hospital } & {8} & {44} & {59} & {39} \\ \hline\end{array}$$

$d f=$ ________

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Problem 50

Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table 11.33.
$$\begin{array}{|l|l|l|l|l|l|}\hline & {20-30} & {30-40} & {40-50} & {50-60} \\ \hline \text { Private Practice } & {16} & {40} & {38} & {6} \\ \hline \text { Hospital } & {8} & {44} & {59} & {39} \\ \hline\end{array}$$

What is the test statistic?

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Problem 51

Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table 11.33.
$$\begin{array}{|l|l|l|l|l|l|}\hline & {20-30} & {30-40} & {40-50} & {50-60} \\ \hline \text { Private Practice } & {16} & {40} & {38} & {6} \\ \hline \text { Hospital } & {8} & {44} & {59} & {39} \\ \hline\end{array}$$

What is the p-value?

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Problem 52

Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table 11.33.
$$\begin{array}{|l|l|l|l|l|l|}\hline & {20-30} & {30-40} & {40-50} & {50-60} \\ \hline \text { Private Practice } & {16} & {40} & {38} & {6} \\ \hline \text { Hospital } & {8} & {44} & {59} & {39} \\ \hline\end{array}$$

What can you conclude at the 5% significance level?

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Problem 53

Which test do you use to decide whether an observed distribution is the same as an expected distribution?

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Problem 54

What is the null hypothesis for the type of test from Exercise 11.53?

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Problem 55

Which test would you use to decide whether two factors have a relationship?

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Problem 56

Which test would you use to decide if two populations have the same distribution?

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Problem 57

How are tests of independence similar to tests for homogeneity?

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Problem 58

How are tests of independence different from tests for homogeneity?

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Problem 59

An archer’s standard deviation for his hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.
What type of test should be used?

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Problem 60

An archer’s standard deviation for his hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.
State the null and alternative hypotheses.

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Problem 61

An archer’s standard deviation for his hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.
Is this a right-tailed, left-tailed, or two-tailed test?

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Problem 62

The standard deviation of heights for students in a school is 0.81. A random sample of 50 students is taken, and the standard deviation of heights of the sample is 0.96. A researcher in charge of the study believes the standard deviation of heights for the school is greater than 0.81.
What type of test should be used?

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Problem 63

The standard deviation of heights for students in a school is 0.81. A random sample of 50 students is taken, and the standard deviation of heights of the sample is 0.96. A researcher in charge of the study believes the standard deviation of heights for the school is greater than 0.81.
State the null and alternative hypotheses.

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Problem 64

The standard deviation of heights for students in a school is 0.81. A random sample of 50 students is taken, and the standard deviation of heights of the sample is 0.96. A researcher in charge of the study believes the standard deviation of heights for the school is greater than 0.81.
$d f=$_______

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Problem 65

The average waiting time in a doctor’s office varies. The standard deviation of waiting times in a doctor’s office is 3.4 minutes. A random sample of 30 patients in the doctor’s office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought.

What type of test should be used?

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Problem 66

The average waiting time in a doctor’s office varies. The standard deviation of waiting times in a doctor’s office is 3.4 minutes. A random sample of 30 patients in the doctor’s office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought.

What is the test statistic?

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Problem 67

The average waiting time in a doctor’s office varies. The standard deviation of waiting times in a doctor’s office is 3.4 minutes. A random sample of 30 patients in the doctor’s office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought.

What is the p-value?

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Problem 68

The average waiting time in a doctor’s office varies. The standard deviation of waiting times in a doctor’s office is 3.4 minutes. A random sample of 30 patients in the doctor’s office has a standard deviation of waiting times of 4.1 minutes. One doctor believes the variance of waiting times is greater than originally thought.

What can you conclude at the 5% significance level?

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Problem 69

Decide whether the following statements are true or false. As the number of degrees of freedom increases, the graph of the chi-square distribution looks more and more symmetrical.

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Problem 70

Decide whether the following statements are true or false. The standard deviation of the chi-square distribution is twice the mean.

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Problem 71

Decide whether the following statements are true or false. The mean and the median of the chi-square distribution are the same if $d f=24$

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Problem 72

A six-sided die is rolled 120 times. Fill in the expected frequency column. Then, conduct a hypothesis test to determine if the die is fair. The data in Table 11.34 are the result of the 120 rolls.

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Problem 73

The marital status distribution of the U.S. male population, ages 15 and older, is as shown in Table 11.35.
$$\begin{array}{|l|l|}\hline \text { Marital Status } & {\text { Percent }} \\ \hline \text { never married } & {31.3} \\ \hline \text { married } & {56.1} \\ \hline \text { widowed } & {2.5} \\ \hline\end{array}$$
$$\begin{array}{|c|c|}\hline \text { Marital Status } & {\text { Percent }} \\ \hline \text { divorced/separated } & {10.1} \\ \hline\end{array}$$

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Problem 74

Perform a goodness-of-fit test to determine whether the local results follow the distribution of the U.S. overall student population based on ethnicity.

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Problem 75

Perform a goodness-of-fit test to determine whether the local results follow the distribution of U.S. AP examinee population, based on ethnicity.

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Problem 76

The City of South Lake Tahoe, CA, has an Asian population of 1,419 people, out of a total population of 23,609. Suppose that a survey of 1,419 self-reported Asians in the Manhattan, NY, area yielded the data in Table 11.38. Conduct a goodness-of-fit test to determine if the self-reported sub-groups of Asians in the Manhattan area fit that of the Lake Tahoe area.

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Problem 77

Conduct a goodness-of-fit test to determine if the actual college majors of graduating females fit the distribution of their expected majors.

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Problem 78

Conduct a goodness-of-fit test to determine if the actual college majors of graduating males fit the distribution of their expected majors.

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Problem 79

In a goodness-of-fit test, the expected values are the values we would expect if the null hypothesis were true.

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Problem 80

In general, if the observed values and expected values of a goodness-of-fit test are not close together, then the test statistic can get very large and on a graph will be way out in the right tail.

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Problem 81

Use a goodness-of-fit test to determine if high school principals believe that students are absent equally during the week or not.

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Problem 82

The test to use to determine if a six-sided die is fair is a goodness-of-fit test.

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Problem 83

In a goodness-of fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.

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Problem 84

A sample of 212 commercial businesses was surveyed for recycling one commodity; a commodity here means any one type of recyclable material such as plastic or aluminum. Table 11.41 shows the business categories in the survey, the sample size of each category, and the number of businesses in each category that recycle one commodity. Based on the study, on average half of the businesses were expected to be recycling one commodity. As a result, the last column shows the expected number of businesses in each category that recycle one commodity. At the 5% significance level, perform a hypothesis test to determine if the observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.

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Problem 85

Table 11.42 contains information from a survey among 499 participants classified according to their age groups. The second column shows the percentage of obese people per age class among the study participants. The last column comes from a different study at the national level that shows the corresponding percentages of obese people in the same age classes in the USA. Perform a hypothesis test at the 5% significance level to determine whether the survey participants are a representative sample of the USA obese population.

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Problem 86

A recent debate about where in the United States skiers believe the skiing is best prompted the following survey. Test to see if the best ski area is independent of the level of the skier.

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Problem 87

Car manufacturers are interested in whether there is a relationship between the size of car an individual drives and the number of people in the driver’s family (that is, whether car size and family size are independent). To test this, suppose that 800 car owners were randomly surveyed with the results in Table 11.44. Conduct a test of independence.

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Problem 88

College students may be interested in whether or not their majors have any effect on starting salaries after graduation. Suppose that 300 recent graduates were surveyed as to their majors in college and their starting salaries after graduation. Table 11.45 shows the data. Conduct a test of independence.

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Problem 89

Some travel agents claim that honeymoon hot spots vary according to age of the bride. Suppose that 280 recent brides were interviewed as to where they spent their honeymoons. The information is given in Table 11.46. Conduct a test of independence.

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Problem 90

A manager of a sports club keeps information concerning the main sport in which members participate and their ages. To test whether there is a relationship between the age of a member and his or her choice of sport, 643 members of the sports club are randomly selected. Conduct a test of independence.

Tony W.
Numerade Educator

Problem 91

A major food manufacturer is concerned that the sales for its skinny french fries have been decreasing. As a part of a feasibility study, the company conducts research into the types of fries sold across the country to determine if the type of fries sold is independent of the area of the country. The results of the study are shown in Table 11.48. Conduct a test of independence.

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Problem 92

According to Dan Lenard, an independent insurance agent in the Buffalo, N.Y. area, the following is a breakdown of the amount of life insurance purchased by males in the following age groups. He is interested in whether the age of the male and the amount of life insurance purchased are independent events. Conduct a test for independence.

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Problem 93

Suppose that 600 thirty-year-olds were surveyed to determine whether or not there is a relationship between the level of education an individual has and salary. Conduct a test of independence.

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Problem 94

The number of degrees of freedom for a test of independence is equal to the sample size minus one.

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Problem 95

The test for independence uses tables of observed and expected data values.

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Problem 96

The test to use when determining if the college or university a student chooses to attend is related to his or her socioeconomic status is a test for independence.

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Problem 97

In a test of independence, the expected number is equal to the row total multiplied by the column total divided by the total surveyed.

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Problem 98

An ice cream maker performs a nationwide survey about favorite flavors of ice cream in different geographic areas of the U.S. Based on Table 11.51, do the numbers suggest that geographic location is independent of favorite ice cream flavors? Test at the 5% significance level.

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Problem 99

Table 11.52 provides a recent survey of the youngest online entrepreneurs whose net worth is estimated at one million dollars or more. Their ages range from 17 to 30. Each cell in the table illustrates the number of entrepreneurs who correspond to the specific age group and their net worth. Are the ages and net worth independent? Perform a test of independence at the 5% significance level.

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Problem 100

A 2013 poll in California surveyed people about taxing sugar-sweetened beverages. The results are presented in Table 11.53, and are classified by ethnic group and response type. Are the poll responses independent of the participants’ ethnic group? Conduct a test of independence at the 5% significance level.

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Problem 101

A psychologist is interested in testing whether there is a difference in the distribution of personality types for business majors and social science majors. The results of the study are shown in Table 11.54. Conduct a test of homogeneity. Test at a 5% level of significance.

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Problem 102

Do men and women select different breakfasts? The breakfasts ordered by randomly selected men and women at a popular breakfast place is shown in Table 11.55. Conduct a test for homogeneity at a 5% level of significance.

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Problem 103

A fisherman is interested in whether the distribution of fish caught in Green Valley Lake is the same as the distribution of fish caught in Echo Lake. Of the 191 randomly selected fish caught in Green Valley Lake, 105 were rainbow trout, 27 were other trout, 35 were bass, and 24 were catfish. Of the 293 randomly selected fish caught in Echo Lake, 115 were rainbow trout, 58 were other trout, 67 were bass, and 53 were catfish. Perform a test for homogeneity at a 5% level of significance.

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Problem 104

In 2007, the United States had 1.5 million homeschooled students, according to the U.S. National Center for Education Statistics. In Table 11.56 you can see that parents decide to homeschool their children for different reasons, and some reasons are ranked by parents as more important than others. According to the survey results shown in the table, is the distribution of applicable reasons the same as the distribution of the most important reason? Provide your assessment at the 5% significance level. Did you expect the result you obtained?

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Problem 105

When looking at energy consumption, we are often interested in detecting trends over time and how they correlate among different countries. The information in Table 11.57 shows the average energy use (in units of kg of oil equivalent per capita) in the USA and the joint European Union countries (EU) for the six-year period 2005 to 2010. Do the energy use values in these two areas come from the same distribution? Perform the analysis at the 5% significance level.

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Problem 106

The Insurance Institute for Highway Safety collects safety information about all types of cars every year, and publishes a report of Top Safety Picks among all cars, makes, and models. Table 11.58 presents the number of Top Safety Picks in six car categories for the two years 2009 and 2013. Analyze the table data to conclude whether the distribution of cars that earned the Top Safety Picks safety award has remained the same between 2009 and 2013. Derive your results at the 5% significance level.

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Problem 107

Is there a difference between the distribution of community college statistics students and the distribution of university statistics students in what technology they use on their homework? Of some randomly selected community college students, 43 used a computer, 102 used a calculator with built in statistics functions, and 65 used a table from the textbook. Of some randomly selected university students, 28 used a computer, 33 used a calculator with built in statistics functions, and 40 used a table from the textbook. Conduct an appropriate hypothesis test using a 0.05 level of significance. Read the statement and decide whether it is true or false.

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Problem 108

If $d f=2,$ the chi-square distribution has a shape that reminds us of the exponential.

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Problem 109

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

Is the traveler disputing the claim about the average or about the variance?

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Problem 110

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

A sample standard deviation of 15 minutes is the same as a sample variance of __________ minutes.

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Problem 111

Is this a right-tailed, left-tailed, or two-tailed test?

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Problem 112

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

$H_{0} :$______

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Problem 113

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

$d f=$______

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Problem 114

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

chi-square test statistic $=$________

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Problem 114

no question

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Problem 115

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

$p$ -value $=$_________

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Problem 116

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade the p-value.

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Problem 117

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

Let $\alpha=0.05$
Decision: ________
Conclusion (write out in a complete sentence.): ________

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Problem 118

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

How did you know to test the variance instead of the mean?

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Problem 119

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

If an additional test were done on the claim of the average delay, which distribution would you use?

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Problem 120

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.

If an additional test were done on the claim of the average delay, but 45 flights were surveyed, which distribution would you use?

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Problem 121

A plant manager is concerned her equipment may need recalibrating. It seems that the actual weight of the 15 oz. cereal boxes it fills has been fluctuating. The standard deviation should be at most 0.5 oz. In order to determine if the machine needs to be recalibrated, 84 randomly selected boxes of cereal from the next day’s production were weighed. The standard deviation of the 84 boxes was 0.54. Does the machine need to be recalibrated?

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Problem 122

Consumers may be interested in whether the cost of a particular calculator varies from store to store. Based on surveying 43 stores, which yielded a sample mean of $\$ 84$ and a sample standard deviation of $\$ 12,$ test the claim that the standard deviation is greater than $\$ 15 .$

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Problem 123

Isabella, an accomplished Bay to Breakers runner, claims that the standard deviation for her time to run the 7.5 mile race is at most three minutes. To test her claim, Rupinder looks up five of her race times. They are 55 minutes, 61 minutes, 58 minutes, 63 minutes, and 57 minutes.

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Problem 124

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. They are also interested in the variation of the number of babies. Suppose that an airline executive believes the average number of babies on flights is six with a variance of nine at most. The airline conducts a survey. The results of the 18 flights surveyed give a sample average of 6.4 with a sample standard deviation of 3.9. Conduct a hypothesis test of the airline executive’s belief.

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Problem 125

The number of births per woman in China is 1.6 down from 5.91 in 1966. This fertility rate has been attributed to the law passed in 1979 restricting births to one per woman. Suppose that a group of students studied whether or not the standard deviation of births per woman was greater than 0.75. They asked 50 women across China the number of births they had had. The results are shown in Table 11.59. Does the students’ survey indicate that the standard deviation is greater than 0.75?

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Problem 126

According to an avid aquarist, the average number of fish in a 20-gallon tank is 10, with a standard deviation of two. His friend, also an aquarist, does not believe that the standard deviation is two. She counts the number of fish in 15 other 20-gallon tanks. Based on the results that follow, do you think that the standard deviation is different from two? Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7; 9; 11; 10; 11

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Problem 127

The manager of "Frenchies" is concerned that patrons are not consistently receiving the same amount of French fries with each order. The chef claims that the standard deviation for a ten-ounce order of fries is at most 1.5 oz., but the manager thinks that it may be higher. He randomly weighs 49 orders of fries, which yields a mean of 11 oz. and a standard deviation of two oz.

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Problem 128

You want to buy a specific computer. A sales representative of the manufacturer claims that retail stores sell this computer at an average price of $\$ 1,249$ with a very narrow standard deviation of $\$ 25 .$ You find a website that has a price comparison for the same computer at a series of stores as follows: $\$ 1,299 ; \$ 1,229.99 ; \$ 1,193.08 ; \$ 1,279 ; \$ 1,224.95$ $\$ 1,229.99 ; \$ 1,269.95 ; \$ 1,249$ Can you argue that pricing has a larger standard deviation than claimed by the manufacturer? Use the 5% significance level. As a potential buyer, what would be the practical conclusion from your analysis?

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Problem 129

A company packages apples by weight. One of the weight grades is Class A apples. Class A apples have a mean weight of 150 g, and there is a maximum allowed weight tolerance of 5% above or below the mean for apples in the same consumer package. A batch of apples is selected to be included in a Class A apple package. Given the following apple weights of the batch, does the fruit comply with the Class A grade weight tolerance requirements. Conduct an appropriate hypothesis test.
(a) at the 5% significance level
(b) at the 1% significance level
Weights in selected apple batch (in grams): 158; 167; 149; 169; 164; 139; 154; 150; 157; 171; 152; 161; 141; 166; 172;

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Problem 130

a. Explain why a goodness-of-fit test and a test of independence are generally right-tailed tests.
b. If you did a left-tailed test, what would you be testing?

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