13. Chi-Square Tests & Goodness of Fit

Finding Expected Frequencies

In Exercises 7–12, (a) calculate the marginal frequencies and (b) find the expected frequency for each cell in the contingency table. Assume that the variables are independent.

Suppose a survey records whether $\mathrm{100}$ students prefer coffee or tea ($\mathrm{Coffee:}60$, $\mathrm{Tea:}40$) and whether they are undergraduate or graduate students ($\mathrm{Undergraduate:}70$, $\mathrm{Graduate:}30$). Which two-way table correctly displays this data?

Which of the following cannot be used to display the relationship between two categorical variables?

Which of the following questions can be answered by using a two-way table?

Conditional Relative Frequencies In Exercises 37–42, use the contingency table from Exercises 33–36, and the information below.

Relative frequencies can also be calculated based on the row totals (by dividing each row entry by the row’s total) or the column totals (by dividing each column entry by the column’s total). These frequencies are conditional relative frequencies and can be used to determine whether an association exists between two categories in a contingency table.

What percent of U.S. adults ages 25 and over who are not high school graduates are unemployed?

Made in America In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, “When you see an ad emphasizing that a product is ‘Made in America,’ are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?” The results of the survey, by age group, are presented in the contingency table below.

d. Construct a conditional distribution for the likelihood of buying a “Made in America” product by age, using age as the explanatory variable.

Contingency Tables and Relative Frequencies In Exercises 33–36, use the information below.

The frequencies in a contingency table can be written as relative frequencies by dividing each frequency by the sample size. The contingency table below shows the number of U.S. adults (in millions) ages 25 and over by employment status and educational attainment. (Adapted from U.S. Census Bureau)

What percent of U.S. adults ages 25 and over (a) are employed and are only high school graduates, (b) are not in the civilian labor force, and (c) are not high school graduates?

Consider the following contingency table, which relates the number of applicants accepted to a college and gender.

e. What proportion of males who applied to the social work school was accepted? What proportion of females who applied to the social work school was accepted?

Happy in Your Marriage? The General Social Survey asks questions about one’s happiness in marriage. Is there an association between gender and happiness in marriage? Use the data in the table to determine if gender is associated with happiness in marriage. Treat gender as the explanatory variable.

Consider the following contingency table, which relates the number of applicants accepted to a college and gender.

a. Construct a conditional distribution of acceptance status by gender.

The table below shows the results from a drug trial for a new ADHD medication. Find the frequencies that we would expect if improvement of symptoms was independent of whether a participant received the placebo.

[DATA] Dropping a Course A survey was conducted at a community college of 50 randomly selected students who dropped a course in the current semester to learn why students drop courses. “Personal” drop reasons include financial, transportation, family issues, health issues, and lack of child care. “Course” drop reasons include reducing one’s load, being unprepared for the course, the course was not what was expected, dissatisfaction with teaching, and not getting the desired grade. “Work” drop reasons include an increase in hours, a change in shift, and obtaining fulltime employment. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_2_17 using the file format of your choice for the version of the text you are using.

a. Construct a contingency table for the two variables.

Made in America In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, “When you see an ad emphasizing that a product is ‘Made in America,’ are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?” The results of the survey, by age group, are presented in the contingency table below.

c. What percentage of Americans are more likely to purchase a product if the advertisement states ""Made in America""?

Consider the following contingency table, which relates the number of applicants accepted to a college and gender.

f. Explain carefully how the bias disappears when type of school is considered.

Dogs Detecting Malaria The following table lists results from an experiment designed to test the ability of dogs to use their extraordinary sense of smell to detect malaria in samples of children’s socks (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Assuming that the dog being correct is independent of whether malaria is present, find the expected value for the observed frequency of 123.