Deck 11: Goodness-Of-Fit and Contingency Tables

Question 1

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are
vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the
flu was recorded. The results are shown below. Use a 0.05 significance level to test the claim that the proportion of people catching the flu is the same in all three groups.

Answer 1

Question 2

Describe a goodness-of-fit test. What assumptions are made when using a goodness-of-fit test?

Answer 2

A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits some
claimed distribution. The assumptions are 1) the sample data are randomly selected; 2) the sample data
consists of frequency counts for the different categories; and 3) for each of the categories, the expected
frequency is at least 5.

Question 3

Use a test to test the claim that in the given contingency table, the row variable and the column variable are
independent. Tests for adverse reactions to a new drug yielded the results given in the table. At the 0.05
significance level, test the claim that the treatment (drug or placebo) is independent of the reaction (whether or
not headaches were experienced).

Answer 3

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Question 4

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or
owned their own home. The results are shown in the table below sorted by gender. At 0, test the claim
that living accommodations are independent of the gender of the student.

Question 5

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is
the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and
asked whether they planned to vote in the next election. The results are shown below.

Question 6

Describe the test of homogeneity. What characteristic distinguishes a test of homogeneity from a test of
independence?

Question 7

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

Question 8

Use the sample data below to test whether car color affects the likelihood of being in an accident. Use a
significance level of 0.01.

Question 9

According to Benford's Law, a variety of different data sets include numbers with leading (first) digits that
follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 18, 0, 79, 476, 180, 8,
23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed
frequencies are substantially different from the frequencies expected with Benford's Law, the check amounts
appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's Law. Does it
appear that the checks are the result of fraud?

Question 10

Explain the computation of expected values for contingency tables in terms of probabilities. Refer to the
assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

Question 11

Perform the indicated goodness-of-fit test. A company manager wishes to test a union leader's claim that
absences occur on the different week days with the same frequencies. Test this claim at the 0.05 level of
significance if the following sample data have been compiled.

Question 12

Among the four northwestern states, Washington has 51% of the total population, Oregon has 30%, Idaho has
11%, and Montana has 8%. A market researcher selects a sample of 1000 subjects, with 450 in Washington, 340
in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance level, test the claim that the sample of 1000
subjects has a distribution that agrees with the distribution of state populations.

Question 13

A researcher wishes to test whether the proportion of college students who smoke is the same at four different
colleges. She randomly selects 100 students from each college and records the number that smoke. The results
are shown below. Use a 0.01 significance level to test the claim that the proportion of students smoking is the same at all four colleges.

Question 14

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results. Use a significance level of 0.05 to test the claim that the die is fair.

Question 15

Use a test to test the claim that in the given contingency table, the row variable and the column variable are
independent. The table below shows the age and favorite type of music of 668 randomly selected people. Use a 5 percent level of significance to test the null hypothesis that age and preferred music type are independent.

Question 16

Use a 2 test to test the claim that in the given contingency table, the row variable and the column variable are
independent. 160 students who were majoring in either math or English were asked a test question, and the
researcher recorded whether they answered the question correctly. The sample results are given below. At the
0.10 significance level, test the claim that response and major are independent.

Question 17

Describe the null hypothesis for the test of independence. List the assumptions for the test of independence.

Question 18

Discuss the three characteristics of a chi-square distribution.

Question 19

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and
chiropractor) to relieve back pain. If we test the claim at a 5% level of significance that success is independent of
the method used, technology provides a P-value of 0.0355. What does the P-value tell us about the claim?

Question 20

The following table shows the number of employees who called in sick at a business for different days of a
particular week. i) At the 0.05 level of significance, test the claim that sick days occur with equal frequency on the different days of the
week.
ii) Test the claim after changing the frequency for Saturday to 152. Describe the effect of this outlier on the test.

Question 21

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or
owned their own home. The results are shown in the table below sorted by gender. At α = 0.05, test the claim
that living accommodations are independent of the gender of the student.

Question 22

According to Benford's Law, a variety of different data sets include numbers with leading (first) digits that
follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 18, 0, 79, 476,
180, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the
observed frequencies are substantially different from the frequencies expected with Benford's Law, the check
amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's
Law. Does it appear that the checks are the result of fraud?

Question 23

Discuss the three characteristics of a chi-square distribution.

Question 24

Describe the null hypothesis for the test of independence. List the assumptions for the test of independence.

Question 25

Describe a goodness-of-fit test. What assumptions are made when using a goodness-of-fit test?

Question 26

Use a 0.01 significance level to test the claim that the proportion of men who plan to vote in the next election is
the same as the proportion of women who plan to vote. 300 men and 300 women were randomly selected and
asked whether they planned to vote in the next election. The results are shown below.

Question 27

Explain the computation of expected values for contingency tables in terms of probabilities. Refer to the
assumptions of the null hypothesis as part of your explanation. You might give a brief example to illustrate.

Question 28

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and
chiropractor) to relieve back pain. If we test the claim at a 5% level of significance that success is independent of
the method used, technology provides a P-value of 0.0655. What does the P-value tell us about the claim?

Question 29

A researcher wishes to test the effectiveness of a flu vaccination. 150 people are vaccinated, 180 people are
vaccinated with a placebo, and 100 people are not vaccinated. The number in each group who later caught the
flu was recorded. The results are shown below.

Question 30

Using the data below and a 0.05 significance level, test the claim that the responses occur with percentages of
15%, 20%, 25%, 25%, and 15% respectively.

Question 31

Use a test to test the claim that in the given contingency table, the row variable and the column variable are
independent. Responses to a survey question are broken down according to employment status and the sample
results are given below. At the 0.10 significance level, test the claim that response and employment status are
independent.

Question 32

Describe the test of homogeneity. What characteristic distinguishes a test of homogeneity from a test of
independence?

Question 33

A researcher wishes to test whether the proportion of college students who smoke is the same in four different
colleges. She randomly selects 100 students from each college and records the number that smoke. The results
are shown below. Use a 0.01 significance level to test the claim that the proportion of students smoking is the same at all four colleges.

Question 34

Define categorical data and give an example.

Question 35

In studying the responses to a multiple-choice test question, the following sample data were obtained. At the
0.05 significance level, test the claim that the responses occur with the same frequency.

Question 36

Use a test to test the claim that in the given contingency table, the row variable and the column variable are
independent. The table below shows the age and favorite type of music of 668 randomly selected people. Use a 5 percent level of significance to test the null hypothesis that age and preferred music type are independent.

Question 37

Use a test to test the claim that in the given contingency table, the row variable and the column variable are
independent. 160 students who were majoring in either math or English were asked a test question, and the
researcher recorded whether they answered the question correctly. The sample results are given below. At the
0.10 significance level, test the claim that response and major are independent.

Question 38

Perform the indicated goodness-of-fit test. Among the four northwestern states, Washington has 51% of the
total population, Oregon has 30%, Idaho has 11%, and Montana has 8%. A market researcher selects a sample of
1000 subjects, with 450 in Washington, 340 in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance
level, test the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state
populations.

Question 39

The table in number 18 is called a two-way table. Why is the terminology of two-way table used?

Question 40

Perform the indicated goodness-of-fit test. You roll a die 48 times with the following results. Use a significance level of 0.05 to test the claim that the die is fair.

Question 41

In studying the occurrence of genetic characteristics, the following sample data were obtained. You would like to test the claim that the characteristics occur with the same frequency at the 0.05 significance level. What is
Value of the test statistic?

Question 42

The following table represents the number of absences on various days of the week at an elementary school.

Question 43

Goodness-of-fit hypothesis tests are always ________.

A) two-tailed
B) right-tailed
C) left-tailed

Question 44

According to Benford's Law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. Test for goodness-of-fit with Benford's Law. When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
Amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 12, 0, 73, 482, 186, 8,
23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed
Frequencies are substantially different from the frequencies expected with Benford's Law, the check amounts
Appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's Law. What is
The value of the test statistic? Does it appear that the checks are the result of fraud?

Question 45

In studying the occurrence of genetic characteristics, the following sample data were obtained. You would like to test the claim that the characteristics occur with the same frequency at the 0.05 significance level. What is the expected value for D?

A) 28
B) 42
C) 38
D) 48

Question 46

While conducting a goodness-of-fit test if the observed and expected values are close, you would expect which of the following:

Question 47

For a recent year, the following are the numbers of homicides that occurred each month in New York City: 38, 30, 46, 40, 46, 49, 47, 50, 50, 42, 37, 37. Use a 0.05 significance level to test the claim that homicides in New York
City are equally likely for each of the 12 months. State your conclusion about the claim.

A) There is not sufficient evidence to warrant rejection of the claim that homicides in New York City are equally likely for each of the 12 months.
B) There is not sufficient evidence to support the claim that that homicides in New York City are equally likely for each of the 12 months.
C) There is sufficient evidence to support the claim that homicides in New York City are equally likely for each of the 12 months.
D) There is sufficient evidence to warrant rejection of the claim that homicides in New York City are equally likely for each of the 12 months.

Question 48

The following are the hypotheses for a test of the claim that college graduation statue and cola preference are independent.

A) Fail to reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that college graduation status and cola preference are independent.
B) Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that college graduation status and cola preference are independent.
C) Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that college graduation status and cola preference are independent.
D) Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that college graduation status and cola preference are independent.

Question 49

True or False: For a test of independence, the population that the data has come from must be normally distributed.

A) True
B) False

Question 50

Select the null hypothesis for a test of independence.

A) The row and column variables are normally distributed.
B) The row and column variables are independent.
C) The row and column variables are dependent.
D) The row and column variables have equal means.

Question 51

Responses to a survey question about color preference for a candy are broken down according to gender in the table given below. At the 0.05 significance level, test the claim that candy color preference and gender are independent. What is your conclusion about the null hypothesis and about the claim?

A) Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
B) Fail to reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
C) Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
D) Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.

Question 52

A survey of students at a college was asked if they lived at home with their parents, rented an apartment, or owned their own home. The results are shown in the table below sorted by gender. At 5, test the claim
That living accommodations are independent of the gender of the student.

A) Reject the null hypothesis. There is sufficient evident to warrant rejection of the claim that living accommodation and gender are independent.
B) Fail to reject the null hypothesis. There is not sufficient evident to warrant rejection of the claim that living accommodation and gender are independent.
C) Reject the null hypothesis. There is not sufficient evident to warrant rejection of the claim that living accommodation and gender are independent.
D) Fail to reject the null hypothesis. There is sufficient evident to warrant rejection of the claim that living accommodation and gender are independent.

Question 53

Which statement is not true for goodness-of-fit tests?

A) Observed frequencies must be whole numbers.
B) The expected frequency is found assuming that the distribution is as claimed.
C) Expected frequencies must be whole numbers.
D) The observed frequency is found from sample data values.

Question 54

A table summarizes the success and failures when subjects used different methods (yoga, acupuncture, and chiropractor) to relieve back pain. If we test the claim at a 5% level of significance that success is independent of
The method used, technology provides a P-value of 0.0355. What does the P-value tell us about the claim?

A) Since the P-value of 0.0355 is lower than 0.05, we reject the null hypothesis of independence between the treatment and whether the subject stops experiencing back pain. This suggests that the choice of treatment
Does appear to make a difference.
B) Since the P-value of 0.0355 is lower than 0.05, we reject the null hypothesis of independence between the treatment and whether the subject stops experiencing back pain. This suggests that the choice of treatment
Does not appear to make a difference.
C) Since the P-value of 0.0355 is greater than 0.05, we reject the null hypothesis of independence between the treatment and whether the subject stops experiencing back pain. This suggests that the choice of treatment
Does appear to make a difference.
D) Since the P-value of 0.0355 is lower than 0.05, we fail to the null hypothesis of independence between the treatment and whether the subject stops experiencing back pain. This suggests that the choice of treatment
Does not appear to make a difference.

Question 55

Which of the following is not a characteristic of a chi-square distribution?

A) As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.
B) The chi-square distribution is different for each number of degrees of freedom.
C) The values of a chi-square distribution cannot be negative.
D) All of the other statements are characteristics of a chi-square distribution.

Question 56

At a high school debate tournament, half of the teams were asked to wear suits and ties and the rest were asked to wear jeans and t-shirts. The results are given in the table below. In order to test the claim at the 0.05 level that
The proportion of wins is the same for teams wearing suits as for teams wearing jeans, what would the null
Hypothesis be?

A) The mean number of wins is different for teams wearing suits as for teams wearing jeans.
B) The proportions of wins is the same for teams wearing suits as for teams wearing jeans.
C) The proportions of wins is different for teams wearing suits as for teams wearing jeans.
D) The mean number of wins is the same for teams wearing suits as for teams wearing jeans.

Question 57

A survey conducted in a small business yielded the results shown in the table. Test the claim that health care coverage is independent of gender. Use a 0.05 significance level. What is the value of the test statistic?

Question 58

The following table represents the number of absences on various days of the week at an elementary school. Identify the number of degrees of freedom for a goodness-of-fit test (for a uniform distribution), assuming a 0.05 significance level.

A) 2
B) 3
C) 4
D) 5

Question 59

At a high school debate tournament, half of the teams were asked to wear suits and ties and the rest were asked to wear jeans and t-shirts. The results are given in the table below. Test the claim at the 0.05 level that the
Proportion of wins is the same for teams wearing suits as for teams wearing jeans. What is your conclusion about the null hypothesis?

A) Fail to support the null hypothesis.
B) Fail to reject the null hypothesis.
C) Reject the null hypothesis.
D) Support the null hypothesis.

Question 60

In conducting a goodness-of-fit test, a requirement is that ________.

A) the observed frequency must be at least ten for each category
B) the observed frequency must be at least five for each category
C) the expected frequency must be at least five for each category
D) the expected frequency must be at least ten for each category

Question 61

Use a significance level of 0.01 to test the claim that workplace accidents are distributed on workdays as follows:
Monday 25%, Tuesday: 15%, Wednesday: 15%, Thursday: 15%, and Friday: 30%.
In a study of 100 workplace accidents, 25 occurred on a Monday, 17 occurred on a Tuesday, 16 occurred on a
Wednesday, 12 occurred on a Thursday, and 30 occurred on a Friday.

Question 62

The following table shows the number of employees who called in sick at a business for different days of a
particular week.

Question 63

An observed frequency distribution of exam scores is as follows:

Question 64

An observed frequency distribution of exam scores is as follows:

Question 65

A company manager wishes to test a union leader's claim that absences occur on the different week days with
the same frequencies. Test this claim at the 0.05 level of significance if the following sample data have been
compiled.

Question 66

You roll a die 48 times with the following results. Use a significance level of 0.05 to test the claim that the die is fair.

Question 67

Responses to a survey question are broken down according to employment status and the sample results are
given below. At the 0.10 significance level, test the claim that response and employment status are independent.

Question 68

In studying the responses to a multiple-choice test question, the following sample data were obtained. At the
0.05 significance level, test the claim that the responses occur with the same frequency.

Question 69

Responses to a survey question are broken down according to gender and the sample results are given below.
At the 0.05 significance level, test the claim that response and gender are independent.

Question 70

When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
amounts from 784 checks issued by seven suspect companies. The frequencies were found to be
0, 9, 0, 70, 485, 189, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9,
respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's
Law, the check amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit
with Benford's Law. Does it appear that the checks are the result of fraud?

Question 71

An observed frequency distribution is as follows:

Question 72

Tests for adverse reactions to a new drug yielded the results given in the table. At the 0.05 significance level, test
the claim that the treatment (drug or placebo) is independent of the reaction (whether or not headaches were
experienced).

Question 73

When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
amounts from 784 checks issued by seven suspect companies. The frequencies were found to be
0, 12, 0, 73, 482, 186, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9,
respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's
Law, the check amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit
with Benford's Law. Does it appear that the checks are the result of fraud?

Question 74

An observed frequency distribution is as follows:

Question 75

Among the four northwestern states, Washington has 51% of the total population, Oregon has 30%, Idaho has
11%, and Montana has 8%. A market researcher selects a sample of 1000 subjects, with 450 in Washington, 340
in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance level, test the claim that the sample of 1000
subjects has a distribution that agrees with the distribution of state populations.

Question 76

Using the data below and a 0.05 significance level, test the claim that the responses occur with percentages of
15%, 20%, 25%, 25%, and 15% respectively.

Question 77

In studying the occurrence of genetic characteristics, the following sample data were obtained. At the 0.05
significance level, test the claim that the characteristics occur with the same frequency.

Question 78

The table below shows the age and favorite type of music of 668 randomly selected people. Use a 5 percent level of significance to test the null hypothesis that age and preferred music type are
independent.

Question 79

160 students who were majoring in either math or English were asked a test question, and the researcher
recorded whether they answered the question correctly. The sample results are given below. At the 0.10
significance level, test the claim that response and major are independent.

Question 80

When working for the Brooklyn District Attorney, investigator Robert Burton analyzed the leading digits of the
amounts from 784 checks issued by seven suspect companies. The frequencies were found to be
0, 18, 0, 79, 476, 180, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9,
respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's
Law, the check amounts appear to result from fraud. Use a 0.05 significance level to test for goodness-of-fit
with Benford's Law. Does it appear that the checks are the result of fraud?