Deck 11: Goodness-Of-Fit and Contingency Tables

Question 1

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.

Answer 1

H₀: The proportion of men who plan to vote in the next election is the same as the proportion of women who plan to vote.
H₁: At least one of the proportions are different.
Test statistic:
Critical value:
Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of 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.

Question 2

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 2

H₀: Treatment and reaction are independent.
H₁: Treatment and reaction are dependent.
Test statistic:
Critical value:
Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that treatment and reaction are independent.

Question 3

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

Answer 3

The test of homogeneity tests the claim that different populations have the same proportions of some
characteristics. In the test of homogeneity, there are predetermined totals for either the rows or columns of the
contingency table. In the test of independence, there is one big sample drawn so that the row and column
totals are determined randomly. In the test of homogeneity, predetermined sample sizes are used for each
population.

Question 4

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

Question 5

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 6

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 7

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

Question 8

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 9

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 10

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 11

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 12

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 13

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

Question 14

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.

Question 15

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 16

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 17

Discuss the three characteristics of a chi-square distribution.

Question 18

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 test the claim that living accommodations are independent of the gender of the student.

Question 19

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 20

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 21

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

Question 22

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 23

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 24

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 25

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 26

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 27

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 28

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

Question 29

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 30

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 test the claim that living accommodations are independent of the gender of the student.

Question 31

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 32

Discuss the three characteristics of a chi-square distribution

Question 33

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 34

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 35

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 36

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 37

Define categorical data and give an example

Question 38

Describe a goodness-of-fit test.

Question 39

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

Question 40

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 41

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 42

The following table represents the number of absences on various days of the week at an elementary school.
Identify the critical value for a goodness-of-fit test, assuming a 0.05 significance level.

A)X²=5.991
B)X²=7.815
C)X²=9.488
D)X²=11.071

Question 43

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?

A) X²=0.1637
B) X²=3.841
C) X²=8.263
D) X²=5.821

Question 44

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 45

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

A) Small X2 value, large P -value
B) Small X² value, small P -value
C) Large X² value, large P -value
D) Large X² value, small P -value

Question 46

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 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.
B) 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.
C) 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.
D) 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.

Question 47

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) 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.
B) Reject the null hypothesis. There is not 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 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 48

The following are the hypotheses for a test of the claim that college graduation statue and cola preference are independent.
H₀: College graduation status and cola preference are independent.
H₁: College graduation status and cola preference are dependent.
If the test statistic: and the critical value is 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 college graduation status and cola preference are independent.
B) 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.
C) Reject the null hypothesis. There is 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 sufficient evidence to warrant rejection of the claim that college graduation status and cola preference are independent.

Question 49

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 proportions of wins is the same for teams wearing suits as for teams wearing jeans.
B) The proportions of wins is different for teams wearing suits as for teams wearing jeans.
C) The mean number of wins is the same 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 50

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

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

Question 51

Goodness-of-fit hypothesis tests are always___________________.

A) Right-tailed
B) Left-tailed
C) Two-tailed

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

Question 53

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 frequenciesare 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?

A) X²=3711.409 It does appear that the checks are the result of fraud.
B) X²=3711.409 It does not appear that the checks are the result of fraud.
C) X²=15.507 It does appear that the checks are the result of fraud.
D) X²=15.507 It does not appear that the checks are the result of fraud.

Question 54

Select the null hypothesis for a test of independence

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

Question 55

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) 38
C) 42
D) 48

Question 56

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

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

Question 57

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?

A) X²=8.263
B) X²=5.991
C) t=8.263
D) X²=5.821

Question 58

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

Question 59

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 theclaim that homicides in New York City are equally likely for each of the 12 months. State your conclusion about the claim.

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

Question 60

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

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