1. Two Categorical Variables: The Chi-Square Test
BPS 7e Chapter 25
© 2015 W. H. Freeman and Company

2. Two-Way Tables
For which of the following pairs of variables would a two-way table of counts be a useful data analysis tool?
age and blood pressure
age and opinion on presidential performance
party affiliation and opinion on presidential performance
diet and growth rate

3. Two-Way Tables (answer)
For which of the following pairs of variables would a two-way table of counts be a useful data analysis tool?
age and blood pressure
age and opinion on presidential performance
party affiliation and opinion on presidential performance
diet and growth rate

4. Two-Way Tables
This table shows the results of a simple random sample asking respondents what types of cars they drive and whether their commuting distance is less than 20 miles. What proportion of the people in the sample drive less than 20 miles to work?
33 / 175 = 0.189
33 / 250 = 0.132
175 / 250 = 0.700
75 / 250 = 0.300
52 / 250 =

5. Two-Way Tables (answer)
This table shows the results of a simple random sample asking respondents what types of cars they drive and whether their commuting distance is less than 20 miles. What proportion of the people in the sample drive less than 20 miles to work?
33 / 175 = 0.189
33 / 250 = 0.132
175 / 250 = 0.700
75 / 250 = 0.300
52 / 250 =

6. Multiple Comparisons
What is the “problem of multiple comparisons” in statistics?
how to sort out the general meaning of a set of comparisons
how to do many comparisons with an overall measure of confidence
how to decide which of a large set of comparisons to do
how to compute a large number of comparisons efficiently

7. Multiple Comparisons (answer)
What is the “problem of multiple comparisons” in statistics?
how to sort out the general meaning of a set of comparisons
how to do many comparisons with an overall measure of confidence
how to decide which of a large set of comparisons to do
how to compute a large number of comparisons efficiently

8. Multiple Comparisons
What is the usual solution to the “problem of multiple comparisons”?
Ignore the problem.
Choose the one most important comparison to actually carry out.
Carry out all comparisons but pay attention only to the one with the largest difference.
Carry out all comparisons but pay attention only to the one with the smallest difference.
Carry out a single overall test followed by a detailed follow-up analysis.

9. Multiple Comparisons (answer)
What is the usual solution to the “problem of multiple comparisons”?
Ignore the problem.
Choose the one most important comparison to actually carry out.
Carry out all comparisons but pay attention only to the one with the largest difference.
Carry out all comparisons but pay attention only to the one with the smallest difference.
Carry out a single overall test followed by a detailed follow-up analysis.

10. Expected Counts Expected counts are calculated assuming: Ha is true.
H0 is true.
neither hypothesis is true.
both hypotheses are true.

11. Expected Counts (answer)
Expected counts are calculated assuming:
Ha is true.
H0 is true.
neither hypothesis is true.
both hypotheses are true.

12. Expected Counts
How is the expected count for a cell of a two-way table calculated?
(row total × column total) / table total
row total / column total
(row total + column total) / table total
(row total / table total) × (column total / table total)

13. Expected Counts (answer)
How is the expected count for a cell of a two-way table calculated?
(row total × column total) / table total
row total / column total
(row total + column total) / table total
(row total / table total) × (column total / table total)

14. Expected Counts
This table shows the results of a simple random sample asking respondents what type of car they drive and whether their commuting distance is less than 20 miles. What is the expected count for the Compact<20 cell?
44 × 61 / 250
61 × 250 / 175
44 × 17 / 250
61 × 175 / 250
( ) / 250

15. Expected Counts (answer)
This table shows the results of a simple random sample asking respondents what type of car they drive and whether their commuting distance is less than 20 miles. What is the expected count for the Compact<20 cell?
44 × 61 / 250
61 × 250 / 175
44 × 17 / 250
61 × 175 / 250
( ) / 250

16. Chi-Square Test Statistic
True or False: The 2 (chi-square statistic) is similar to all other test statistics we have discussed in that it compares observed values with values that would be expected if H0 were true.
True
False

17. Chi-Square Test Statistic (answer)
True or False: The 2 (chi-square statistic) is similar to all other test statistics we have discussed in that it compares observed values with values that would be expected if H0 were true.
True
False

18. Chi-Square Test Statistic
Which statement is NOT correct about the 2 test statistic?
The test statistic is the sum of positive numbers and therefore must be positive.
A small value of the test statistic would indicate evidence supporting the null hypothesis.
A large value of the test statistic would be in support of the alternative hypothesis.
A value close to 0 would indicate that expected counts are much different from observed counts.

19. Chi-Square Test Statistic (answer)
Which statement is NOT correct about the 2 test statistic?
The test statistic is the sum of positive numbers and therefore must be positive.
A small value of the test statistic would indicate evidence supporting the null hypothesis.
A large value of the test statistic would be in support of the alternative hypothesis.
A value close to 0 would indicate that expected counts are much different from observed counts.

20. Chi-Square Test Statistic
Which of the following is the formula for the chi-square statistic?

21. Chi-Square Test Statistic (answer)
Which of the following is the formula for the chi-square statistic?

22. Chi-Square Test Statistic
True or False: In calculating the chi-square, each term in the sum represents the “contribution” to the chi-square statistic for one of the cells in the table. In this regard, even rounding each term to two decimal places can still create round-off error in the sum.
True
False

23. Chi-Square Test Statistic (answer)
True or False: In calculating the chi-square, each term in the sum represents the “contribution” to the chi-square statistic for one of the cells in the table. In this regard, even rounding each term to two decimal places can still create round-off error in the sum.
True
False

24. Chi-Square Test Statistic
We can safely use the chi-square test with critical values from the chi-square distribution when no more than _____ of the expected counts are less than ___ and all individual expected counts are ___ or greater.
20%; 1; 5
15%; 1; 5
20%; 5; 1
15%; 5; 1

25. Chi-Square Test Statistic (answer)
We can safely use the chi-square test with critical values from the chi-square distribution when no more than _____ of the expected counts are less than ___ and all individual expected counts are ___ or greater.
20%; 1; 5
15%; 1; 5
20%; 5; 1
15%; 5; 1

26. Cell Counts Required
Here is a two-way table of counts. The expected counts are in red. The data are based on an SRS with two categorical variables. Is it safe to use the chi-square test with critical values from the chi-square distribution?
yes, because all of the observed counts are 5 or greater
yes, because all of the expected counts are 5 or greater
no, because the expected counts are not whole numbers
yes, because the sum of observed counts = sum of expected 50
51.35 36
31.65 12
14.99 16
18.34 13
11.31 6
5.36 54
50.30 25
31.01 17
14.69

27. Cell Counts Required (answer)
Here is a two-way table of counts. The expected counts are in red. The data are based on an SRS with two categorical variables. Is it safe to use the chi-square test with critical values from the chi-square distribution?
yes, because all of the observed counts are 5 or greater
yes, because all of the expected counts are 5 or greater
no, because the expected counts are not whole numbers
yes, because the sum of observed counts = sum of expected 50
51.35 36
31.65 12
14.99 16
18.34 13
11.31 6
5.36 54
50.30 25
31.01 17
14.69

28. Chi-Square Tests
Using a computer program, the chi-square test statistic for the commuter data is 2 = 5.37 with P-value < If we use  = 0.05, what conclusion is appropriate?
Don’t reject H0; the results are significant.
Don’t reject H0; the results are not significant.
Reject H0; the results are significant.
Reject H0; the results are not significant.

29. Chi-Square Tests (answer)
Using a computer program, the chi-square test statistic for the commuter data is 2 = 5.37 with P-value < If we use  = 0.05, what conclusion is appropriate?
Don’t reject H0; the results are significant.
Don’t reject H0; the results are not significant.
Reject H0; the results are significant.
Reject H0; the results are not significant.

30. Chi-Square Tests
If the chi-square test is significant, what should we do as a follow-up?
Compare expected counts with observed counts in each cell.
Examine the components of the chi-square test for each cell.
Informally compare the conditional distributions.
All of the above

31. Chi-Square Tests (answer)
If the chi-square test is significant, what should we do as a follow-up?
Compare expected counts with observed counts in each cell.
Examine the components of the chi-square test for each cell.
Informally compare the conditional distributions.
All of the above

32. Chi-Square Tests
Using a computer program, the chi-square test statistic for the commuter data is 2 = 15.2 with P-value < If we use  = 0.05, what conclusion is appropriate?
Don’t reject H0; the results are significant.
Don’t reject H0; the results are not significant.
Reject H0; the results are significant.
Reject H0; the results are not significant.

33. Chi-Square Tests (answer)
Using a computer program, the chi-square test statistic for the commuter data is 2 = 15.2 with P-value < If we use  = 0.05, what conclusion is appropriate?
Don’t reject H0; the results are significant.
Don’t reject H0; the results are not significant.
Reject H0; the results are significant.
Reject H0; the results are not significant.

34. Uses of the Chi-Square Test
Students in a simple random sample of sixth-graders were asked which of three choices is most important to them: making good grades, being popular, or participating in sports. Also recorded was the students’ gender.
What type of data-collection situation was this?
independent SRSs with individuals classified by one categorical variable
single SRS with individuals classified by two categorical variables

35. Uses of the Chi-Square Test (answer)
Students in a simple random sample of sixth-graders were asked which of three choices is most important to them: making good grades, being popular, or participating in sports. Also recorded was the students’ gender.
What type of data-collection situation was this?
independent SRSs with individuals classified by one categorical variable
single SRS with individuals classified by two categorical variables

36. Uses of the Chi-Square Test
What are the hypotheses for the chi-square test for two-way tables of counts?
H0: no relationship, Ha: relationship
H0: relationship, Ha: no relationship
H0: strong relationship, Ha: weak relationship

37. Uses of the Chi-Square Test (answer)
What are the hypotheses for the chi-square test for two-way tables of counts?
H0: no relationship, Ha: relationship
H0: relationship, Ha: no relationship
H0: strong relationship, Ha: weak relationship

38. Uses of the Chi-Square Test
Students in a simple random sample of sixth-graders were asked which of three choices is most important to them: making good grades, being popular, or participating in sports. Also recorded was the students’ gender.
The chi-square test is used to test the null hypothesis:
H0: no linear relationship between gender and choice.
H0: no relationship between gender and choice.
H0: no difference between the means of gender and choice.

39. Uses of the Chi-Square Test (answer)
Students in a simple random sample of sixth-graders were asked which of three choices is most important to them: making good grades, being popular, or participating in sports. Also recorded was the students’ gender.
The chi-square test is used to test the null hypothesis:
H0: no linear relationship between gender and choice.
H0: no relationship between gender and choice.
H0: no difference between the means of gender and choice.

40. Uses of the Chi-Square Test
To see if percentages of high school graduates in seven states in New England are equal, researchers took samples from each state and recorded whether the respondent was a high school graduate.
What type of data-collection system was this?
independent SRSs with individuals classified by one categorical variable
single SRS with individuals classified by two categorical variables

41. Uses of the Chi-Square Test (answer)
To see if percentages of high school graduates in seven states in New England are equal, researchers took samples from each state and recorded whether the respondent was a high school graduate.
What type of data-collection system was this?
independent SRSs with individuals classified by one categorical variable
single SRS with individuals classified by two categorical variables

42. Uses of the Chi-Square Test
When police respond to spousal-abuse calls, they can either arrest the offender, issue a citation to the offender, and/or separate the couple. A group of researchers studied whether one method was better than the others in deterring further instances of spousal abuse. When presented with an eligible case, dispatchers randomly selected the method the officers should use when responding to the call. The following data were collected:
What type of data-collection system was this?
independent SRSs with individuals classified by one categorical variable
single SRS with individuals classified by two categorical variables

43. Uses of the Chi-Square Test (answer)
When police respond to spousal-abuse calls, they can either arrest the offender, issue a citation to the offender, and/or separate the couple. A group of researchers studied whether one method was better than the others in deterring further instances of spousal abuse. When presented with an eligible case, dispatchers randomly selected the method the officers should use when responding to the call. The following data were collected:
What type of data-collection system was this?
independent SRSs with individuals classified by one categorical variable
single SRS with individuals classified by two categorical variables

44. Chi-Square Distributions
Which statement is NOT correct about the 2 distribution?
The area under the curve is equal to 1, or 100%.
It has a left-skewed shape.
The values are always positive.
It has different curves depending on the degrees of freedom.

45. Chi-Square Distributions (answer)
Which statement is NOT correct about the 2 distribution?
The area under the curve is equal to 1, or 100%.
It has a left-skewed shape.
The values are always positive.
It has different curves depending on the degrees of freedom.

46. Chi-Square Distributions
A researcher wanted to see if there was a relationship between education-attainment level and whether someone smokes or does not smoke. She classified education attainment into five categories—did not graduate from high school, high school graduate, some college, college graduate, or graduate degree—and then performed a chi-square test on the data. What were the degrees of freedom?
(5) (2) = 10
(4) (2) = 8
(4) (1) = 4
It cannot be determined without the data.

47. Chi-Square Distributions (answer)
A researcher wanted to see if there was a relationship between education-attainment level and whether someone smokes or does not smoke. She classified education attainment into five categories—did not graduate from high school, high school graduate, some college, college graduate, or graduate degree—and then performed a chi-square test on the data. What were the degrees of freedom?
(5) (2) = 10
(4) (2) = 8
(4) (1) = 4
It cannot be determined without the data.