Title: Chi-Square Tests
Course: Question Banks

Q: Which sample will have a larger chi-square test statistic? (See if you can tell without doing any calculations.)
- Sample #4 (In both samples only two colors have a non-zero chi-square term. Sample #4 blue and orange have an absolute error of 10. Similarly, in Sample #3 red and yellow both have an absolute error of 10. Which has a larger relative error?)
- Sample #3 and Sample #4 have the same chi-square value. (In both samples only two colors have a non-zero chi-square term. Did you focus on absolute error? Sample #4 blue and orange have an absolute error of 10. Similarly, in Sample #3 red and yellow both have an absolute error of 10. But the chi-square test statistic is based on relative error. Which has a larger relative error?)
- Sample #3 (The chi-square value for Sample #3 is 100/36 + 100/42, which is about 4.82. The chi-square value for Sample #4 is 100/72 + 100/60, which is about 3.06.)
A: Sample #3

Q: What can we say about the value of χ^2 if the sample distribution matches the null distribution exactly?
- the χ^2 value is less than 0 (The chi-square value can never be negative. Each term in the calculation of the chi-square statistic is positive since we square the differences between observed and expected counts.)
- the χ^2 value is greater than 0 (If the sample distribution matches the null distribution exactly, then the observed count equals the expected count for each category, which means the difference is zero. So the chi-square statistic is zero.)
- none of the above; we need to know the observed and expected values. (If the sample distribution matches the null distribution exactly, then the observed count equals the expected count for each category. We do not need to know the observed and expected values because we know the difference will be 0 for each term in the chi-square calculation.)
- the χ^2 value is 0 (If the sample distribution matches the null distribution exactly, then the observed count equals the expected count for each category, which means the difference is zero. So the chi-square statistic is zero.)
A: the χ^2 value is 0

Q: A company in the United States produces packs of Pokemon cards. According to the company, 45% of the cards feature normal-type Pokemon, 25% feature fire-type Pokemon, 20% feature water-type Pokemon, and 10% feature electric-type Pokemon. In a pack of 40 cards, what is the expected count of water-type Pokemon?
- 4 (We expect 20% of the sample to feature water-type Pokemon and 20% of 40 is not 4.)
- It is impossible to tell because samples of 40 cards will vary. (It is of course true that the samples will vary. But the expected counts do not vary. The expected counts are what we expect to see if the null hypothesis is true. We expect 20% of the sample to feature water-type Pokemon. 20% of 40 is 8.)
- 8 (We expect 20% of the sample to feature water-type Pokemon. 20% of 40 is 8.)
A: 8

Q: Suppose that we conduct a study of college students to investigate the relationship between gender and body piercing. Here is the data.

Which comparison of conditional percents is relevant to our investigation?
- 576/648 , 64/352, 640/1000 (None of these percents are relevant. 648/1000 is the probability that a student in the sample is pierced. The other two percents should represent the probability that a female student (or male student) is pierced since gender is the explanatory variable.)
- 576/1000, 72/1000, 648/1000 (The first two percents here are joint percents, not conditional percents. Which variable is the explanatory variable? We want to compare conditional percents where the denominator comes from the explanatory variable.)
- 576/640, 72/360, 648/1000 (648/1000 is the probability that a student in the sample is pierced. The other two percents represent the probability that a female student (or male student) is pierced. This is relevant to our investigation since gender is the explanatory variable.)
A: 576/640, 72/360, 648/1000

Q: Suppose that we conduct a study of college students to investigate the relationship between gender and body piercing. The null hypothesis says that the two variables are independent (there is no relationship.) Here is the data.

What is the expected count of non-pierced females?
- 414.72 (It seems like you found the expected count of pierced females. Read the question and try again.)
- 126.72 (It seems like you found the expected count of non-pierced males. Read the question and try again.)
- 64 (64 is the observed count of non-pierced females, and we are looking for the expected count.)
- 225.28 (We expect 352/1000 (35.2%) of the students to not be pierced. If the piercing and gender are independent, then 35.2% of the females in the sample will not be pierced. 0.352(640) = 225.28)
A: 225.28

Q: A researcher surveys a random sample of 750 college students about their political views. The researcher’s survey contains questions about the following variables:

- Age
- Political Affiliation (Democrat / Republican / Other)
- Annual income
- number of hours worked per week
- Residential status (on-campus / off-campus)

The researcher will conduct a chi-square test of independence. Which pair of variables can the researcher use?
- age and political affiliation (The chi-square test for independence looks for a relationship between two categorical variables. Political affiliation is a categorical variable, but age is a quantitative variable.)
- age and annual income (The chi-square test for independence looks for a relationship between two categorical variables. Both age and annual income are quantitative variables.)
- annual income and number of hours worked per week (The chi-square test for independence looks for a relationship between two categorical variables. Both number of hours worked per week and annual income are quantitative variables.)
- residential status and number of hours worked per week (The chi-square test for independence looks for a relationship between two categorical variables. Residential status is a categorical variable, but number of hours per week is a quantitative variable.)
- political affiliation and residential status (The chi-square test for independence looks for a relationship between two categorical variables. Both political affiliation and residential status are categorical variables.)
A: political affiliation and residential status

Q: Suppose that you are a researcher conducting a survey about the 2016 presidential election. You survey a random sample of people to find out how they voted. Here are the results:
Which chi-square test is appropriate for this data (assuming the conditions are met)?
- Goodness-of-fit (The goodness-of-fit test is for a one-way table.)
- Test of homogeneity (You are right that the test of homogeneity is for a two-way table. But with a test of homogeneity we should select independent random samples from each population or subgroup. Here the data comes from a single random sample.)
- Test of independence (The test of independence is for a two-way table. In addition, with a test of independence we select one random sample from the population and collect data for two categorical variables.)
A: Test of independence

Q: A researcher wants to find out which type of pet is the most popular. She conducts a survey that asks this question of a random sample of U.S. adults in 2010 and a second independent sample in 2017. She then conducts a chi-square test of homogeneity to determine if there are statistically significant differences in the distribution of responses for these two years.

In this results table, the observed count appears above the expected count in each cell.

If we included an exploratory data analysis with the test of homogeneity, the percentages most appropriate as part of this analysis for the Dog group are ________ .
- 193/517 compared to 324/517 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. We want to compare the conditional percentages for pet preference for the two years. Note that the denominators should be from the explanatory variable (year). The option you chose are conditional percentages where the denominator is from the response variable (pet preference).)
- 193/1998 compared to 324/1998 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. We want to compare the conditional percentages for pet preference for the two years. The option you chose are joint percentages from the combined samples. Note that the denominators should be from the explanatory variable (year).)
- 193/1517 compared to 994/1998 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. We want to compare the conditional percentages for pet preference for the two years. The option you chose compares adults with a dog preference in the 2010 group to a marginal percentage.)
- 193/994 compared to 324/1004 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. These ratios give the conditional percentages for dog preference for the two years. Note that the denominators are from the explanatory variable (year).)
A: 193/994 compared to 324/1004

Q: A researcher wants to find out which type of pet is the most popular. She conducts a survey that asks this question of a random sample of U.S. adults in 2010 and a second independent sample in 2017. She then conducts a chi-square test of homogeneity to determine if there are statistically significant differences in the distribution of responses for these two years.

In this results table, the observed count appears above the expected count in each cell.

If we included an exploratory data analysis with the test of homogeneity, the percentages most appropriate as part of this analysis for the Cat group are ________ .
- 466/892 compared to 426/892 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. We want to compare the conditional percentages for pet preference for the two years. Note that the denominators should be from the explanatory variable (year). The option you chose are conditional percentages where the denominator is from the response variable (pet preference).)
- 466/1998 compared to 426/1998 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. We want to compare the conditional percentages for pet preference for the two years. The option you chose are joint percentages from the combined samples. Note that the denominators should be from the explanatory variable (year).)
- 466/892 compared to 994/1998 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. We want to compare the conditional percentages for pet preference for the two years. The option you chose compares adults with a cat preference in the 2010 group to a marginal percentage.)
- 466/944 compared to 426/1004 (In a test of homogeneity we compare the distribution of the response variable across the comparison groups. These ratios give the conditional percentages for cat preference for the two years. Note that the denominators are from the explanatory variable (year).)
A: 466/944 compared to 426/1004