Deck 17: The Chi-Square Statistic: Tests for Goodness of Fit and Independence

Question 1

Which of the following accurately describes the chi-square test for goodness of fit?​

A)​It is similar to a single-sample t test because it uses one sample to test a hypothesis about one population.
B)​It is similar to a correlation because it uses one sample to evaluate the relationship between two variables.
C)​It is similar to an independent-measures t test because it uses separate samples to evaluate the difference between separate populations.
D)​It is similar to both a correlation and an independent-measures t test because it can be used to evaluate a relationship between variables or a difference between populations.

Answer 1

​It is similar to a single-sample t test because it uses one sample to test a hypothesis about one population.

Question 2

What happens to the critical value for a chi-square test if the size of the sample is increased?​

A)​The critical value also increases.
B)​The critical value decreases.
C)​The critical value depends on the number of categories, not the sample size.
D)​The critical value is determined entirely by the alpha level.

Answer 2

​The critical value depends on the number of categories, not the sample size.

Question 3

Which of the following accurately describes the observed frequencies for a chi-square test?​

A)​They are always whole numbers.
B)​They can contain fractions or decimal values.
C)​They can contain both positive and negative values.
D)​They are always the same value.

Answer 3

​They are always whole numbers.

Question 4

For a given alpha level,what happens to the critical value for a chi-square test if the number of categories is increased?​

A)​The critical value also increases.
B)​The critical value decreases.
C)​The critical value depends on the sample size, not the number of categories.
D)​The critical value is determined entirely by the alpha level.

Question 5

The term expected frequencies refers to the frequencies ____.​

A)​found in the sample data
B)​found in the population being examined
C)​computed from the null hypothesis
D)​that are hypothesized for the population being examined

Question 6

How does the difference between f_e and f_o influence the outcome of a chi-square test?​

A)​the larger the difference, the larger the value of chi-square, and the greater the likelihood of rejecting the null hypothesis
B)​the larger the difference, the larger the value of chi-square, and the lower the likelihood of rejecting the null hypothesis
C)​the larger the difference, the smaller the value of chi-square, and the greater the likelihood of rejecting the null hypothesis
D)​the larger the difference, the smaller the value of chi-square, and the lower the likelihood of rejecting the null hypothesis

Question 7

A researcher obtains a negative value for a chi-square statistic.What can you conclude because the value is negative?​

A)​The observed frequencies are consistently larger than the expected frequencies.
B)​The expected frequencies are consistently larger than the observed frequencies.
C)​There are large differences between the observed and expected frequencies.
D)​The researcher made a mistake; the value of the chi-square cannot be negative.

Question 8

For a fixed level of significance,the critical value for the chi-square will ____.​

A)​increase as the df value increases
B)​decrease as the df value increases
C)​increase as the sample size increases
D)​decrease as the sample size increases

Question 9

​Ten years ago,only 15% of the U.S.population consisted of people more than 65 years old.A researcher plans to use a sample of n = 300 people to determine whether the population distribution has changed during the past ten years.If a chi-square test is used to evaluate the data,what is the expected frequency for the older-than-65 category?

A)​15
B)​30
C)​45
D)​150

Question 10

Which of the following best describes the possible values for a chi-square statistic?​

A)​Chi-square is always a positive whole number.
B)​Chi-square is always positive, but can contain fractions or decimal values.
C)​Chi-square can be either positive or negative, but always is a whole number.
D)​Chi-square can be either positive or negative and can contain fractions or decimals.

Question 11

A chi-square test for goodness of fit has df = 2.How many categories were used to classify the individuals in the sample?​

A)​two
B)​three
C)​four
D)​five

Question 12

What is evaluated by the chi-square test for goodness of fit?​

A)​the relationship between two variables
B)​the mean differences between two or more treatments
C)​the shape or proportions for a population distribution
D)​the significance of a regression estimate

Question 13

The term observed frequencies refers to the frequencies ____.​

A)​found in the sample data
B)​found in the population being examined
C)​computed from the null hypothesis
D)​that are hypothesized for the population being examined

Question 14

A researcher used a sample of n = 60 individuals to determine whether there are any preferences among four brands of pizza.Each individual tastes all four brands and selects his/her favorite.If the data are evaluated with a chi-square test for goodness of fit using α = .05,then how large does the chi-square statistic need to be to reject the null hypothesis?​

A)​greater than 7.81
B)​less than 7.81
C)​less than 5.99
D)between 5.99 and 7.81​

Question 15

​Which of the following accurately describes the chi-square distribution?

A)​symmetrical with a mean of zero
B)​positively skewed with all values greater than or equal to zero
C)​negatively skewed with all values greater than or equal to zero
D)​symmetrical with a mean equal to n - 1

Question 16

A sample of n = 100 people is classified into four categories.If the results are evaluated with a chi-square test for goodness of fit,what is the df value for the chi-square statistic?​

A)​3
B)​4
C)​99
D)​100

Question 17

What conclusion is appropriate if a chi-square test produces a chi-square statistic near zero?​

A)There is a good fit between the sample data and the null hypothesis.​
B)​There is a large discrepancy between the sample data and the null hypothesis.
C)​All of the expected frequencies must also be close to zero.
D)​The researcher made a mistake because chi-square can never be close to zero.

Question 18

What values are specified by the null hypothesis for the chi-square test for goodness of fit?​

A)​proportions for the entire population
B)​proportions for the sample
C)​frequencies for the entire population
D)​frequencies for the sample

Question 19

What is indicated by a large value for the chi-square statistic?​

A)​ The sample data (observed values) do not match the null hypothesis.
B) ​There is a good fit between the sample data (observed values) and the null hypothesis.
C)​ There is a good fit between the sample data (observed values) and the null hypothesis.
D)​ The observed values from the sample data are consistently smaller than the expected values.

Question 20

Which of the following accurately describes the expected frequencies for a chi-square test?​

A)​They are always whole numbers.
B)​They can contain fractions or decimal values.
C)They can contain both positive and negative values.​
D)​They are positive whole numbers.

Question 21

A chi-square test for independence has df = 2.What is the total number of categories (cells in the matrix)that were used to classify individuals in the sample?​

A)​two
B)​three
C)​four
D)​six

Question 22

A chi-square test for independence is being used to evaluate the relationship between two variables,one of which is classified into three categories and the second of which is classified into four categories.The chi-square statistic for this test would have a df value equal to ____.​

A)​6
B)​7
C)​10
D)​11

Question 23

Which of the following is a basic assumption for a chi-square hypothesis test?​

A)​The population distribution(s) must be normal.
B)​The scores must come from an interval or ratio scale.
C)​The observations must be independent.
D)​The sample size is less than 30.

Question 24

A chi-square test for independence has df = 3.What is the total number of categories (cells in the matrix)that were used to classify individuals in the sample?​

A)​four
B)​six
C)​eight
D)​nine

Question 25

Which of the following accurately describes the chi-square test for independence?​

A)​It is similar to a single-sample t test because it uses one sample to test a hypothesis about one population.
B)​It is similar to a correlation because it uses one sample to evaluate the relationship between two variables.
C)​It is similar to an independent-measures t test because it uses separate samples to evaluate the difference between separate populations.
D)​It is similar to both a correlation and an independent-measures t test because it can be used to evaluate a relationship between variables or a difference between populations.

Question 26

A chi-square test for independence is used to evaluate the relationship between two variables.If both variables are classified into two categories,then what is the df value for the chi-square statistic?​

A)​df = 1
B)​df = 2
C)​df = 3
D)​The value of df cannot be determined from the information provided.

Question 27

Under what circumstances will the chi-square test for goodness of fit produce a large value for a chi-square?​

A)​when the sample proportions match the hypothesized population proportions
B)​when the sample proportions are much different than the hypothesized population proportions
C)​when the sample mean is close to the population mean
D)​when there is a large difference between the sample mean and the population mean

Question 28

A researcher selects a sample of 100 people to investigate the relationship between gender (male/female)and registering to vote.The sample consists of 40 females,of whom 30 are registered voters,and 60 males,of whom 40 are registered voters.If these data are used for a chi-square test for independence,what is the expected frequency for registered females?

A)​12
B)​28
C)​40
D)​42

Question 29

Under what circumstances should the chi-square statistic not be used?​

A)​if f_e > 5 for any cell
B)​if f_e < 5 for any cell
C)​if f_e = f_o for any cell
D)​if f_e =7 for any cell

Question 30

When the data form a 2 x 2 matrix,the phi-coefficient is used to measure effect size for the chi-square test for independence.If other factors are held constant,how does the sample size influence the values for phi and chi-square?​

A)​A larger sample increases both the values of phi and chi-square.
B)​A larger sample increases the value of phi but has no effect on the value of chi-square.
C)​A larger sample increases the value of chi-square but has no effect on the value of phi.
D)​Sample size does not influence either the value of phi or chi-square.

Question 31

A chi-square test for goodness of fit is used to examine the distribution of individuals across three categories,and a chi-square test for independence is used to examine the distribution of individuals in a 2 X 2 matrix of categories.Which test has the larger value for df?​

A)​The test for goodness of fit has the larger value.
B)​The test for independence has the larger value.
C)​Both tests have the same df value.
D)​The df value depends on the sizes of the samples that are used.

Question 32

How do you compute the df value for the chi-square test for independence?​

A)​df = n - 1
B)​df = n - 2
C)​df = n - C
D)​df = (R-1)(C-1)

Question 33

A sample of 100 people is classified by gender (male/female)and by whether or not they are registered voters.The sample consists of 80 females and 20 males,and has a total of 60 registered voters.If these data were used for a chi-square test for independence,what is the expected frequency for males who are registered voters?​

A)​12
B)​20
C)​40
D)​60

Question 34

A researcher selects a sample of 100 people to investigate the relationship between gender (male/female)and registering to vote.The sample consists of 40 females,of whom 30 are registered voters,and 60 males,of whom 40 are registered voters.If these data are used for a chi-square test for independence,what is the observed frequency for registered males?

A)​12
B)​28
C)​40
D)​42

Question 35

What is stated by the null hypothesis for the chi-square test for independence?​

A)​There is a relationship between the two variables.
B)​There is no relationship between the two variables.
C)​Both variables have the same frequency distribution.
D)​The two variables have different frequency distributions.

Question 36

A chi-square test for goodness of fit is used to examine the distribution of individuals across four categories,and a chi-square test for independence is used to examine the distribution of individuals across six categories in a 2 X 3 matrix of categories.Which test has the larger value for df?​

A)​The goodness of fit has the larger value.
B)​The test for independence has the larger value.
C)​Both tests have the same df value.
D)​The df value depends on the sizes of the samples that are used.

Question 37

In a chi-square test for independence or goodness of fit,____.​

A)​Σf_e = n
B)​Σf_e = Σf_o
C)​both Σf_e = n and Σf_e = Σf_o
D)​neither Σf_e = n nor Σf_e = Σf_o

Question 38

Under what conditions is Cramér's V used to measure effect size for a chi-square test for independence?​

A)​when both variables consist of exactly two categories
B)​when both variables consist of more than two categories
C)​when either of the two variables consists of exactly two categories
D)​when either of the two variables consists of more than two categories

Question 39

​A researcher is conducting a chi-square test for independence to evaluate the relationship between gender and preference among three different designs for a new automobile.Each individual in a sample of n = 30 males and n = 30 females selects a favorite design from the three choices.If the researcher obtains a chi-square statistic of χ² = 4.81,what is the appropriate statistical decision for the test?

A)​Reject the null hypothesis with α = .05 but not with α = .01.
B)​Reject the null hypothesis with either α = .05 or α = .01.
C)​Fail to reject the null hypothesis with either α = .05 or α = .01.
D)​Fail to reject the null hypothesis with α = .10.

Question 40

A sample of 100 people is classified by gender (male/female)and by whether or not they are registered voters.The sample consists of 80 females and 20 males,and has a total of 60 registered voters.If these data are used for a chi-square test for independence,what is the total number of females for the expected frequencies?​

A)​32
B)​20
C)​48
D)​80

Question 41

For a fixed level of significance,the critical value for chi-square increases as the size of the sample increases.​

Question 42

The chi-square test for goodness of fit evaluates the distribution for one variable,and the test for independence evaluates the relationship between two variables.​

Question 43

The chi-square statistic measures how well the data fit the hypothesis,and the symbol is χ².​

Question 44

For a fixed level of significance,the critical value for chi-square increases as the degrees of freedom increase.​

Question 45

A chi-square test for independence produces a chi-square statistic with df = 2.The data for this research study form a matrix with four separate cells.​

Question 46

For the chi-square test for goodness-of-fit,the researcher must select a sample with an equal number of individuals in each category.​

Question 47

The chi-square test for independence uses the same formula for the chi-square statistic as the chi-square test for goodness-of-fit.​

Question 48

A large value for the chi-square statistic indicates a large discrepancy between the sample data and the hypothesis.​

Question 49

The observed frequencies for a chi-square test are always whole numbers (no fractions or decimals).​

Question 50

The chi-square test for independence is similar to an independent-measures t test because it can be used to evaluate the difference between two populations or two treatment conditions.​

Question 51

For a chi-square test,the expected frequencies are calculated values that are intended to produce a sample that is representative of the null hypothesis.​

Question 52

For a chi-square test,the observed frequencies are obtained from the sample.​

Question 53

Tests that concern parameters and require assumptions about parameters are called parametric tests.​

Question 54

The data for a chi-square test for independence can be viewed either as representing one sample with two measurements for each participant or as two (or more)separate samples.​

Question 55

One characteristic of nonparametric tests is that they make few,if any,assumptions about the populations being investigated.​

Question 56

​One characteristic of the chi-square tests is that they can be used when the data are measured on a nominal scale.

Question 57

The chi-square distribution tends to be symmetrical with a mean of μ = 0.​

Question 58

For both the chi-square test for goodness of fit and the chi-square for independence,Σf_e = n.​

Question 59

In general,a large value for chi-square indicates a good fit between the sample data and the null hypothesis.​

Question 60

The chi-square test for independence is similar to a correlation because it evaluates the relationship between two variables.​

Question 61

A researcher is interested consumer preferences among three brands of yogurt.A sample of 90 people is obtained and each individual is asked to taste each brand and then select his/her favorite.The resulting frequency data are as follows:​ ​Do the data indicate any significant preferences among the three brands? Test at the .05 level of significance.

Question 62

If other factors are held constant,increasing the sample size for a chi-square test for independence will increase the likelihood of rejecting the null hypothesis.​

Question 63

The phi-coefficient is used to evaluate effect size for a chi-square test for goodness of fit provided there are only two categories.​

Question 64

For a chi-square test for independence,the expected frequencies for any row are exactly the same as the expected frequencies in every other row.​

Question 65

An insurance investigator has observed the people with some astrological signs tend to be safer drivers than people with other signs.Using insurance records,the investigator classified 200 people according to their astrological signs and whether or not they were involved in a car accident during the previous 12 months.The data are as follows: ​Do the data indicate a significant relationship between sign and accidents? Test at the .05 level of significance.

Question 66

Explain how the chi-square tests differ from parametric tests (such as t tests or ANOVA)with respect to the hypotheses,the data,and the assumptions underlying the test.

Question 67

A researcher would like to know whether there is a consistent,predictable relationship between verbal skills and math skills for high school students.A sample of 200 students is obtained and each student is given a standardized English test and a standardized math test.Based on the test results,students are classified as high or low for verbal skills and for math skills.The results are summarized in the following frequency distribution: ​
a. Based on these results, can the researcher conclude that there is a significant relationship between verbal skills and math skills? Test at the .05 level of significance.
b. Compute the phi-coefficient to measure the strength of the relationship.

Question 68

Cramér's V is used to evaluate effect size for the chi-square test for independence whenever the test has df > 1.​

Question 69

Although the outcome for a chi-square test for independence is influenced by sample size,measures of effect size such as the phi-coefficient or Cramér's V are not.​

Question 70

The primary purpose of the chi-square test and the tests for mean differences is to evaluate the strength of a relationship,while the primary purpose of a Pearson correlation is to measure the significance of the relationship.​

Question 71

A chi-square test should not be used if any expected frequency is less than five.​

Question 72

A researcher is interested in the relationship between IQ and longevity.A sample of n = 200 obituaries is selected from the newspaper and old school records are consulted to determine the individuals' IQ scores when they were between the ages of 10 and 20.The individuals were then classified into two IQ groups and three age groups at the time of death.The data are as follows.Based on these results,is there a significant relationship between IQ and longevity? Test with α = .05.​

Question 73

Cramér's V is used to measure effect size for the chi-square test for independence and is superior to Cohen's w because it makes allowances for the size of the data matrix.​

Question 74

Cohen's w is a measure of effect size that can be used for both the chi-square test for independence and for goodness of fit.

Question 75

The phi-coefficient is used to evaluate effect size for the chi-square test for independence,provided the test has df = 1.​