Deck 15: Chi-Squared Tests

Question 1

If the expected frequency of a cell is less than 5, you should increase the significance level.

Answer 1

False

Question 2

A multinomial experiment with two categories is identical to a binomial experiment.

Answer 2

True

Question 3

If there are only two categories, the chi-squared goodness-of-fit test is the same as the z -test for p , the population proportion (as long as the sample\cell sizes meet the conditions).

Answer 3

True

Question 4

For a chi-squared distributed random variable with 10 degrees of freedom and a level of significance of 0.025, the chi-squared table value is 20.4831. Suppose the value of your test statistic is 16.857. This will lead you to reject the null hypothesis.

Question 5

A small chi-squared test statistic in a goodness-of-fit test supports the null hypothesis.

Question 6

If the expected frequency of a cell is less than 5, you should combine cells of the table.

Question 7

The only way the chi-squared test statistic can be zero is if the observed frequencies are all exactly the same as the expected frequencies.

Question 8

A chi-squared test is used to describe a population of nominal data.

Question 9

You cannot use a chi-squared goodness-of-fit test when there are only two possible outcomes for each trial in your experiment.

Question 10

The total of the observed frequencies in a multinomial experiment must equal nk where n is the number of trials and k is the number of categories.

Question 11

A chi-squared distribution is symmetric.

Question 12

A left-tailed area in the chi-squared distribution equals 0.90. For 10 degrees of freedom the table value equals 15.9871.

Question 13

The alternative hypothesis in a goodness-of-fit test is that none of the p_i values are equal to their values specified in H ₀.

Question 14

In conducting a chi-squared goodness-of-fit test, an essential condition is that all expected frequencies are at least five.

Question 15

All of the expected frequencies in a chi-squared goodness-of-fit test must be equal to each other.

Question 16

A chi-squared goodness-of-fit test is always a two-tailed test.

Question 17

In a goodness-of-fit test, all of the proportions specified in the null hypothesis must be equal to each other.

Question 18

For a chi-squared distributed random variable with 12 degrees of freedom and a level of significance of 0.05, the test statistics is 25.168. The chi-squared value from the table is 21.0261. These results will lead us to reject the null hypothesis.

Question 19

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

Question 20

If the observed frequencies are all smaller than the expected frequencies in a goodness-of-fit test, the chi-squared test statistic will be negative.

Question 21

Suppose the value of your chi-squared test statistic in a goodness-of-fit test is equal to 0. What do you conclude?

A)Reject H ₀. Conclude that at least one proportion is not equal to its specified value.
B)Fail to reject H ₀. Not enough evidence to say the proportions are different from what is listed in H ₀.
C)Not enough information; need the degrees of freedom for the test.
D)None of these choices.

Question 22

A chi-squared test of a contingency table with 6 degrees of freedom results in a test statistic of 13.25. Using the chi-squared table, the most accurate statement that can be made about the p -value for this test is that p -value is greater than 0.025 but smaller than 0.05.

Question 23

The degrees of freedom for the test statistic in a test of a contingency table is ( r - 1 )( c - 1 )where r is the number of rows in the table, and c is the number of columns.

Question 24

To calculate the expected values in a test of a contingency table, you assume that the null hypothesis is true.

Question 25

In a test of a contingency table, rejecting the null hypothesis concludes the variables are not independent.

Question 26

In a chi-squared test of a contingency table, the value of the test statistic was c ² = 15.652, and the critical value at a = 0.025 was 11.1433. Thus, we must reject the null hypothesis at a = 0.025.

Question 27

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

Question 28

If each element in a population is classified into one and only one of several categories, the population is:

A)normal.
B)multinomial.
C)chi-squared.
D)None of these choices.

Question 29

In a goodness-of-fit test, H ₀ lists specific values for proportions and the test of a contingency table does not.

Question 30

To determine whether data were drawn from a multinomial distribution with certain proportions, you use a:

A)chi-squared goodness-of-fit test.
B)chi-squared test of a contingency table.
C)chi-squared test for normality.
D)None of these choices.

Question 31

A chi-squared test of a contingency table is applied to a contingency table with 4 rows and 4 columns for two qualitative variables. The degrees of freedom for this test must be 9.

Question 32

The test statistic for the chi-squared test of a contingency table is the same as the test statistic for the goodness-of-fit test.

Question 33

If two events A and B are independent, the P ( A and B )= P ( A )+ P ( B ).

Question 34

To determine the critical values in the chi-squared distribution table, you need to know the:

A)sample size.
B)degrees of freedom
C)probability of Type II error.
D)All of these choices are true.

Question 35

A chi-squared goodness-of-fit test is always conducted as a(n):

A)lower-tail test.
B)upper-tail test.
C)two-tail test.
D)All of these choices are true.

Question 36

A chi-squared test of a contingency table is applied to a contingency table with 3 rows and 4 columns for two qualitative variables. The degrees of freedom for this test must be 12.

Question 37

In the test of a contingency table, the observed cell frequencies must satisfy the rule of 5.

Question 38

To produce expected values for a test of a contingency table, you multiply estimated joint probabilities for each cell by the total sample size, n .

Question 39

The expected frequency for the cell in row i and column j is the row i total plus the row j total, all divided by n .

Question 40

In the test of a contingency table, the expected cell frequencies must satisfy the rule of 5.

Question 41

Which of the following statements regarding the chi-squared distribution is true?

A)The chi-squared distribution is skewed to the right.
B)All values of the chi-squared distribution are greater than or equal to zero.
C)The critical region for a goodness-of-fit test with k categories is c ² > c ² _{a , k -1}, where c ² is the value of the test statistic.
D)All of these choices are true.

Question 42

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

A)0
B)0.05
C)1.96
D)45

Question 43

Consider a multinomial experiment with 200 trials, where the outcome of each trial is classified into one of 5 categories. The number of degrees of freedom associated with the chi-squared goodness-of-fit test equals:

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

Question 44

How does a multinomial distribution differ from a binomial distribution?

A)A binomial has only two possible categories and a multinomial can have more.
B)A binomial has a fixed number of n trials. A multinomial has a fixed number of nk trials, where k is the number of categories.
C)The probabilities in a binomial distribution are always p and 1 - p . The trials in a multinomial distribution are always p\k and (1 - p\k ).
D)All of these choices are true.

Question 45

A left tail area in the chi-squared distribution equals 0.99. For df = 8, the table value equals:

A)20.090
B)3.4895
C)2.7326
D)15.5073

Question 46

The chi-squared test of a contingency table is based upon:

A)one quantitative variable.
B)two quantitative variables.
C)one qualitative variable.
D)two qualitative variables.

Question 47

The rule of five requires that the:

A)observed frequency for each cell must be at least 5.
B)degrees of freedom for the test must be at least 5.
C)expected frequency for each cell must be at least 5.
D)difference between the observed and expected frequency for each cell must be at least 5.

Question 48

How do you find the probabilities needed to obtain expected frequencies for a test of a contingency table?

A)If there are r rows and c columns, let each probability be 1\ rc .
B)Use the probabilities specified in the null hypothesis.
C)Assume H ₀ is true and use your data to calculate them.
D)None of these choices.

Question 49

To address whether two variables are related in a contingency table, the alternative hypothesis, H ₁, is:

A)The two variables are independent.
B)The two variables are dependent.
C)The two variables are equal.
D)None of these choices.

Question 50

The sampling distribution of the test statistic for a goodness-of-fit test with k categories is a:

A)chi-squared distribution with k - 1 degrees of freedom.
B)normal distribution.
C)Student t -distribution with k - 1 degrees of freedom.
D)None of these choices.

Question 51

If you reject H ₀ in a test of a contingency table, you conclude that based on your data:

A)The two nominal variables are independent.
B)The two nominal variables are equal.
C)The two nominal variables have the same proportions listed in H ₀.
D)None of these choices.

Question 52

Which of the following represents H ₁ in a chi-squared goodness-of-fit test to see if all 5 colors of a certain candy appear in the same proportion in the population?

A)H ₁: p ₁ = p ₂ = p ₃ = p ₄ = p ₅ = 0.20.
B)H ₁: At least one proportion is not equal to 0.20.
C)H ₁: None of these proportions are equal.
D)None of these choices.

Question 53

How do you calculate the expected frequency for one cell in a goodness-of-fit test?

A)The expected frequency is equal to the proportion specified in H ₀ for that cell.
B)Use the total number of observations divided by the number of categories.
C)Multiply the specified proportion for that cell (found in H ₀)by the total sample size.
D)None of these choices.

Question 54

Which of the following conditions indicate that H ₀ should be rejected in a goodness-of-fit test?

A)The observed frequencies are equal to their expected frequencies.
B)The test statistic is large.
C)The degrees of freedom is large.
D)All of these choices are true.

Question 55

The number of degrees of freedom for a contingency table with r rows and c columns is:

A)r + c
B)rc
C)( r - 1 )( c - 1 )
D)None of these choices.

Question 56

If we use the c ² goodness-of-fit to test for the differences among 4 proportions, the degrees of freedom is equal to:

A)3
B)4
C)5
D)None of these choices.

Question 57

If the expected frequency e_i for any cell i is less than 5, we should:

A)choose another sample with five or more observations.
B)use the normal distribution instead of the chi-squared distribution.
C)combine cells such that each observed frequency f_i   is 5 or more.
D)increase the degrees of freedom by 5.

Question 58

Contingency tables are used in:

A)testing independence of two samples.
B)testing dependence in matched pairs.
C)testing independence of two qualitative variables in a population.
D)describing a single population.

Question 59

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

A)The experiment consists of a fixed number of trials.
B)The outcome of each trial is classified into one of two possible categories.
C)The probability p_i that the outcome will fall into cell i remains constant for each trial.
D)Each trial of the experiment is independent of the other trials.

Question 60

To address whether two variables are related in a contingency table, the null hypothesis, H ₀, says that

A)The two variables are independent.
B)The two variables are dependent.
C)The two variables are equal.
D)None of these choices.

Question 61

You find the expected value for each cell in a chi-squared test for a contingency table by multiplying the ____________________ total by the ____________________ total and dividing by the sample size, n .

Question 62

The rule of ____________________ states that in order to conduct the chi-squared goodness-of-fit test, the expected value for each cell must be ____________________ or more.

Question 63

A chi-squared test statistic in a test of a contingency table that is equal to zero means:

A)The two nominal variables are equal.
B)The two nominal variables are independent.
C)The two nominal variables have the same proportions listed in H ₀.
D)All of these choices.

Question 64

A chi-squared distribution has a shape that is ____________________.

Question 65

The null hypothesis in a chi-squared test of a contingency table is that the two nominal variables are ____________________.

Question 66

The values of a chi-squared distribution are always ____________________ zero.

Question 67

A chi-squared test of a contingency table with 4 rows and 5 columns shows that the value of the test statistic is 22.18. Using a chi-squared table, the most accurate statement that can be made is:

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 68

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

A)The chi-squared test of a multinomial experiment.
B)The chi-squared 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 69

The chi-squared test of a(n)____________________ is used to determine whether there is enough evidence to say two nominal variables are related.

Question 70

A chi-squared test of a contingency table with 6 degrees of freedom results in a test statistic c ² = 13.58. Using the c ² tables, the most accurate statement that can be made about the p -value for this test is that:

A)p -value > 0.10
B)p -value > 0.05
C)0.05 p -value
D)0.025 p -value

Question 71

The alternative hypothesis in a chi-squared test of a contingency table is that the two nominal variables are ____________________.

Question 72

The rule of five states that in order to conduct the chi-squared goodness-of-fit test, the ____________________ value for each cell must be five or more.

Question 73

The number of degrees of freedom for a contingency table with 4 rows and 8 columns is

A)32
B)12
C)21
D)10

Question 74

When k = 2 the ____________________ experiment is identical to the ____________________ experiment.

Question 75

If the expected frequencies and the observed frequencies are quite different, you are likely to ____________________ the null hypothesis.

Question 76

A test statistic that lies in the far right tail of the chi-squared distribution indicates you will ____________________ H ₀.

Question 77

The alternative hypothesis of a goodness-of-fit test states that ____________________ of the proportions is not equal to its value specified in H ₀.

Question 78

The chi-squared goodness-of-fit test compares the ____________________ frequencies in the table to the ____________________ frequencies based on the null hypothesis.

Question 79

A large chi-squared test statistic in a test of a contingency table means you conclude:

A)The two nominal variables are dependent.
B)The two nominal variables are equal.
C)The two nominal variables have the same proportions listed in H ₀.
D)None of these choices.

Question 80

A(n)____________________ experiment is like a binomial experiment except it contains two or more categories.