Deck 14: Analysis of Categorical Data

Question 1

Of the values for a chi-square test statistic listed below, which one is likely to lead to rejecting the null hypothesis in a goodness-of-fit test?

A) 0
B) 1.3
C) 1.9
D) 40

Answer 1

D

Question 2

In chi-square tests, there is a rule known as Rule of Five. What does this rule require?

A) It requires that the observed frequency for each cell be five or more.
B) It requires that the degrees of freedom for the test be at least five.
C) It requires that the expected frequency for each cell be five or more.
D) It requires that the difference between the observed and expected frequency for each cell be at least five.

Answer 2

C

Question 3

In a chi-square goodness-of-fit test with 5 degrees of freedom and a significance level of 0.05, the chi-square value from the table is 11.0705. Which of the following computed values of the chi-square test statistic will lead to rejection of the null hypothesis?

A) 7.814
B) 8.952
C) 10.78
D) 17.61

Answer 3

D

Question 4

Which of the following must be known in order to determine the critical values in the chi-square distribution table?

A) the degrees of freedom
B) the probability of Type I error
C) the probability of Type II error
D) both the degrees of freedom and the probability of Type I error

Question 5

Which of the following is the appropriate test to use if we wish to determine whether there is evidence that the proportion of successes is higher in group 1 than in group 2?

A) z test
B) test
C) t test with 2 degrees of freedom
D) F test with 2 degrees of freedom

Question 6

In a goodness-of-fit test, suppose that a sample showed that the observed frequency and the expected frequency were equal for each cell i. Using this information, what may one conclude about the null hypothesis?

A) It is rejected at0.05 but is not rejected at0.025.
B) It is not rejected at0.05 but is rejected at0.025.
C) It is rejected at any level.
D) It is not rejected at anylevel.

Question 7

In the chi-square goodness-of-fit test, if the expected frequencies, , and the observed frequencies, , were very different, what would we conclude?

A) The null hypothesis is false, and we would reject it.
B) The null hypothesis is true, and we would not reject it.
C) The alternative hypothesis is false, and we would reject it.
D) The chi-square distribution is invalid, and we would use the t-distribution instead.

Question 8

How is a chi-square goodness-of-fit test conducted?

A) as a lower-tailed test
B) as an upper-tailed test
C) as a two-tailed test
D) either as a lower-tailed or as an upper-tailed test

Question 9

Consider a multinomial experiment with 200 trials, and the outcome of each trial can be classified into one of five categories. In this case, how many degrees of freedom would be associated with the chi-square goodness-of-fit test?

A) 195
B) 40
C) 5
D) 4

Question 10

To determine whether a single coin is fair, the coin was tossed 250 times, and heads was observed 140 times. What is the value of the test statistic?

A) 13.6
B) 30
C) 40
D) 110

Question 11

Which of the following best describes the sampling distribution of the test statistic for a goodness-of-fit test with k categories?

A) It is a Student t distribution with k - 1 degrees of freedom.
B) It is a normal distribution.
C) It is a chi-square distribution with k - 1 degrees of freedom.
D) It is an approximately chi-square distribution with k - 1 degrees of freedom.

Question 12

Which of the following is NOT a characteristic of a multinomial experiment?

A) The experiment consists of a fixed number, n, of trials.
B) The outcome of each trial can be classified into one of two categories called successes and failures.
C) The probabilitythat the outcome will fall into cell i remain constant for each trial.
D) Each trial of the experiment is independent of the other trials.

Question 13

In a goodness-of-fit test, suppose that the value of the test statistic is 11.89 and the degrees of freedom are 5. At the 5% significance level, what may one conclude about the null hypothesis?

A) It is rejected, and the p-value for the test is smaller than 0.05.
B) It is not rejected, and the p-value for the test is greater than 0.05.
C) It is rejected, and the p-value for the test is greater than 0.05.
D) It is not rejected, and the p-value for the test is smaller than 0.05.

Question 14

If each element in a population is classified into one and only one of several categories, which of the following best describes this kind of population?

A) It is a normal population.
B) It is a multinomial population.
C) It is a chi-square population.
D) It is a binomial population.

Question 15

In a goodness-of-fit test, suppose that the value of the test statistic is 13.08 and the number of degrees of freedom is 6. At the 5% significance level, what may one conclude about the null hypothesis?

A) It is rejected, and the p-value for the test is smaller than 0.05.
B) It is not rejected, and the p-value for the test is greater than 0.05.
C) It is rejected, and the p-value for the test is greater than 0.05.
D) It is not rejected, and the p-value for the test is smaller than 0.05.

Question 16

A left-tailed area in the chi-square distribution equals 0.90. Taking this into consideration, for 7 degrees of freedom, what does the table value equal?

A) 1.68987
B) 2.83311
C) 12.0170
D) 14.0671

Question 17

Which statistical technique is appropriate when we describe a single population of qualitative data with two or more categories?

A) the z test of the difference between two proportions
B) the chi-square goodness-of-fit test
C) the chi-square test of a contingency table
D) either (a) and (b)

Question 18

The area to the right of a chi-square value is 0.05. Given this information, for 9 degrees of freedom, what would be the table value?

A) 3.32511
B) 4.16816
C) 16.9190
D) 19.0228

Question 19

Which of the following is our best option if the expected frequency for any cell i is less than five and we want to use Pearson's chi-square statistic in our experiment?

A) We must choose another sample of five or more observations.
B) We should use the normal distribution instead of the chi-square distribution.
C) We should combine the cells such that each observed frequencyis five or more.
D) We increase the number of degrees of freedom for the test by five.

Question 20

Consider a cell in a contingency table. Given the cell's row total of 80, the cell's column total of 60, and a sample size of 250, what is the cell's expected frequency?

A) 1.786
B) 3.125
C) 19.2
D) 20.0

Question 21

If we wanted to conduct a two-tailed test of a population proportion, which of these tests could we employ?

A) z test of a population proportion
B) the chi-square goodness-of-fit test since
C) the chi-square test of independence
D) both a and b

Question 22

What summary data does a contingency table contain?

A) It classifies data with respect to two qualitative variables.
B) It divides each variable into two or more categories.
C) It contains numbers that show the frequency of occurrence of all possible combinations of categories.
D) All of (a), (b), and (c).

Question 23

What is the appropriate test to use if we wish to determine whether there is evidence that the proportion of successes is the same in group 1 as in group 2?

A) the z test
B) thetest
C) the z test and thetest
D) t test with 2 degrees of freedom

Question 24

Which statistical technique is appropriate when we wish to analyze the relationship between two qualitative variables with two or more categories?

A) the chi-square test of a multinomial experiment
B) the chi-square test of a contingency table
C) the t test of the difference between two means
D) the z test of the difference between two proportions

Question 25

Consider a cell in a contingency table. Given the cell's row total of 200, the cell's column total of 75, and a sample size of 1000, what is the cell's expected frequency?

A) 10
B) 15
C) 20
D) 44

Question 26

Which of the following statements is NOT a characteristic of a goodness-of-fit test?

A) It determines the likelihood that sample data have been generated from a population that conforms to a specified type of probability distribution.
B) It compares the entire shapes of two (discrete or continuous) probability distributions: one describing known population data and the other one describing hypothetical sample data.
C) The aim of the test might be limited to identifying only the family to which the underlying distribution belongs.
D) The aim of the test might be limited to identifying only the family to which the underlying distribution belongs or it might go further, seeking even to identify a particular member of that family.

Question 27

Which of the following statements is NOT a property of a contingency table test?

A) The nature of the sampling distribution of chi-square depends on the number of degrees of freedom associated with the under investigation.
B) The degrees of freedom are found as (r - 2)(c - 2), where r represents number of rows and c represents number of columns.
C) The contingency table must have a minimum of two rows and two columns.

Question 28

If we wanted to conduct a one-tailed test of a population proportion, which of the following tests could we employ?

A) z test of a population proportion
B) the chi-square goodness-of-fit test since
C) the chi-square test of independence

Question 29

A chi-square test of a contingency table with 4 rows and 5 columns shows that the value of the test statistic is 22.18. Which of the following is the most accurate statement that can be made about the p-value for this test?

A) p-value is greater than 0.05
B) p-value is smaller than 0.025
C) p-value is greater than 0.025 but smaller than 0.05
D) p-value is greater than 0.10

Question 30

How many columns would the table associated with a contingency table test with 4 rows and 15 degrees of freedom have?

A) 5 columns
B) 6 columns
C) 9 columns
D) 11 columns

Question 31

Which of the following procedures would be considered a suitable application for the chi-square technique?

A) testing the alleged independence of two qualitative variables
B) making inferences about the relative sizes of more than two population proportions
C) conducting a goodness-of-fit-test to determine whether data are consistent with data drawn from a particular probability distribution

Question 32

In a chi-square test of independence, the value of the test statistic was , and the critical value at was 11.1433. Which of the following conclusions may we draw from this information?

A) We fail to reject the null hypothesis at.
B) We reject the null hypothesis at.
C) We don't have enough evidence to accept or reject the null hypothesis at.
D) We should decrease the level of significance in order to reject the null hypothesis.

Question 33

The president of a large university collected data from students concerning building a new library, and classified the responses into different categories (strongly agree, agree, undecided, disagree, strongly disagree) and according to whether the student was male or female. To determine whether the data provide sufficient evidence to indicate that the responses depend upon gender, which of the following would be the most appropriate test?

A) a chi-square goodness-of-fit test
B) a chi-square test of a contingency table (test of independence)
C) a chi-square test of normality
D) a chi-square test of abnormality

Question 34

A chi-square test of independence is applied to a contingency table with four rows and five columns for two qualitative variables. What are the degrees of freedom for this test?

A) 20
B) 16
C) 15
D) 12

Question 35

Upon which of the following types of variables is the chi-square test of a contingency table based?

A) two qualitative variables
B) two quantitative variables
C) three or more qualitative variables
D) three or more quantitative variables

Question 36

How many columns does the table associated with a contingency table test with 4 rows and 30 degrees of freedom have?

A) 7 columns
B) 9 columns
C) 11 columns
D) 13 columns

Question 37

Upon which of the following kinds of variables is the chi-square test of independence based?

A) two qualitative variables
B) two quantitative variables
C) three or more qualitative variables
D) three or more quantitative variables

Question 38

Which of the following is the number of degrees of freedom for a contingency table with six rows and six columns?

A) 36
B) 25
C) 12
D) 6

Question 39

A chi-square test of independence with 10 degrees of freedom results in a test statistic of 19.25. Using the chi-square table, which of the following is the most accurate statement that can be made about the p-value for this test?

A) p-value < 0.025
B) 0.025 < p-value < 0.05
C) 0.05 < p-value < 0.10
D) 0.10 < p-value < 0.20

Question 40

In which of the following situations are contingency tables used?

A) to test independence of two samples
B) to test dependence in matched pairs
C) to test independence of two qualitative variables in a population
D) to describe a single population

Question 41

The rejection region of the chi-square goodness-of-fit test has k - 1 degrees of freedom, where k is the number of categories (called cells).

Question 42

A multinomial experiment, where the outcome of each trial can be classified into one of two categories, is identical to a binomial experiment.

Question 43

A right-tailed area in the chi-square distribution equals 0.05. For 6 degrees of freedom the table value equals 12.5916.

Question 44

A chi-square goodness-of-fit test with 3 degrees of freedom results in a test statistic of 6.789. Using the chi-square table, the most accurate statement that can be made about the p-value for this test is that 0.05 < p-value < 0.10.

Question 45

If the test statistic for a chi-square goodness-of-fit test is larger than the critical value, the null hypothesis should be rejected.

Question 46

Whenever the expected frequency of a cell is less than five, one remedy for this condition is to increase the significance level.

Question 47

A left-tailed area in the chi-square distribution equals 0.10. For 5 degrees of freedom the table value equals 9.23635.

Question 48

A goodness-of-fit test determines the likelihood that sample data have been generated from a population that conforms to a specified type of probability distribution.

Question 49

The rejection region of the chi-square goodness-of-fit test is , where k is the number of categories and is the value of the test statistics.

Question 50

A chi-square goodness-of-fit test is always conducted as a two-tailed test.

Question 51

For a chi-square distributed random variable with 10 degrees of freedom and a level of significance of 0.025, the chi-square table value is 20.4831. The computed value of the test statistics is 16.857. This will lead us to reject the null hypothesis.

Question 52

The chi-square goodness-of-fit test involves two categorical variables.

Question 53

The degrees of freedom associated with a goodness-of-fit test equal the number of rows times the number of columns in the table.

Question 54

The middle 0.95 portion of the chi-square distribution with 9 degrees of freedom has table values of 3.32511 and 16.9190, respectively.

Question 55

When the expected cell frequencies are smaller than five, the cells should be combined in a meaningful way such that the expected cell frequencies exceed five.

Question 56

Whenever the expected frequency of a cell is less than five, one remedy for this condition is to increase the size of the sample.

Question 57

Whenever the expected frequency of a cell is less than five, one remedy for this condition is to decrease the size of the sample.

Question 58

The area to the right of a chi-square value is 0.01. For 8 degrees of freedom, the table value is 1.64648.

Question 59

In applying the chi-square goodness-of-fit test, the rule of thumb for all expected frequencies is that each expected frequency is five or more.

Question 60

A chi-square goodness-of-fit test is always conducted as a two-tailed test.

Question 61

Chi-square tests of independence are always lower-tailed because a perfect fit between and makes the test statistic equal to 0.

Question 62

The degrees of freedom associated with a chi-square test of independence where data are summarized in a contingency table with r rows and c columns equal the number of rows times the number of columns in the table; that is, rc.

Question 63

In a typical chi-square test of independence, we calculate each expected cell frequency ( ) as the product of row total ( ) times column total ( ) times sample size (n).

Question 64

By combining cells that have expected frequencies smaller than five, we guard against having an inflated test statistic that could have led us to incorrectly accept the null hypothesis.

Question 65

In a typical contingency table test, we calculate each expected cell frequency as the product of row total times column total times sample size.

Question 66

The degrees of freedom associated with a contingency table test of independence equal the number of rows times the number of columns in the table.

Question 67

A contingency table classifies data according to two or more categories associated with each of two qualitative variables that are statistically independent of one another.

Question 68

Numbers in a contingency table show the frequency of occurrence of all possible combinations of categories.

Question 69

A chi-square test for independence is applied to a contingency table with four rows and four columns for two qualitative variables. The degrees of freedom for this test must be 9.

Question 70

A contingency table classifies data with respect to two qualitative variables that are each divided into two or more categories.

Question 71

To be valid, a chi-square test of independence requires that each expected frequency equal 30 or more.

Question 72

The chi-square test statistic for a contingency table with r rows and c columns can be negative if r is much smaller than
c.

Question 73

To be valid, a chi-square test of independence requires that the expected frequency for each cell in the contingency table equals ten or more.

Question 74

In a chi-square test of independence with 6 degrees of freedom and a level of significance of 0.05, the critical value from the chi-square table is 12.5916. The computed value of the test statistics is 11.264. This will lead us to reject the null hypothesis.

Question 75

A chi-square test for independence is applied to a contingency table with three rows and five columns for two qualitative variables. The degrees of freedom for this test is 8.

Question 76

A chi-square test for independence is applied to a contingency table with three rows and four columns for two qualitative variables. The degrees of freedom for this test must be 12.

Question 77

In a chi-square test of independence, the value of the test statistic was = 15.652, and the critical value at was 11.1433. Thus, we must reject the null hypothesis at .

Question 78

Numbers in a contingency table show the frequency of occurrence of all possible combinations of categories.

Question 79

Chi-square tests of independence are always lower-tailed because a perfect fit between and makes equal to 0.

Question 80

A chi-square test for independence with 10 degrees of freedom results in a test statistic of 17.894. Using the chi-square table, the most accurate statement that can be made about the p-value for this test is that 0.05 < p-value < 0.10.