Analyzing Categorical Data Practice Questions

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Analyzing Categorical Data Practice Questions

Practice

A small sandwich shop has six different sandwiches on the menu: turkey, beef, veggie, salami, ham, and pastrami. Unfortunately, the shop has been losing business and, to save money, has to discontinue one of its sandwiches. The owner of the shop believes that 28%, 12%, 20%, 14%, 22%, and 4% of his customers order turkey, beef, veggie, salami, ham, and pastrami, respectively. He plans a study to determine whether or not these percentages are correct.

1. Give the response variable and identify the parameters associated with its categories.
2. Identify the hypotheses to be tested.

Find the p-values associated with the following values of the test statistic, x²:

3. x² = 9.348, df= 3
4. x² = 9.348, df = 5
5. x² = 35, df = 11
6. x² = 9.348, df = 10

The sandwich shop owner randomly selected 150 of his customers that purchased a sandwich within the last month and recorded the type of sandwich the customer ordered. Of the 150 orders, 45 ordered turkey, 14 ordered beef, 33 ordered veggie, 16 ordered salami, 37 ordered ham, and 5 ordered pastrami.

7. Find the expected counts within each sandwich category.
8. Determine the value of the test statistic.
9. Find the p-value associated with the test statistic.
10. Determine whether or not to reject the null hypothesis and state the conclusion in the context of the problem.

A researcher wanted to know whether men and women drive the same types of vehicles in the researcher's home state. She randomly selected 125 males and 125 females from all vehicle owners in her state and recorded what type of vehicle each drives. The data are presented in Table 20.6.

11. State the null and alternative hypotheses of interest to this researcher.
12. Find the expected counts.
13. Verify whether or not the conditions for the test are satisfied and, if so, find the value of the test statistic.
14. Find the p-value.
15. Decide whether or not to reject the null hypothesis.
16. State the conclusions in the context of the problem.
17. The student government wanted to know whether or not the freshmen who live on campus develop more school spirit by the end of the academic year than those who do not live on campus. They commissioned a poll to find out. The pollster randomly selected freshmen to participate in the study. Each selected freshman was asked whether or not he or she lives on campus and asked whether he or she considers him- or herself to have no school spirit, some school spirit, or a lot of school spirit. The results are shown in Table 20.9.

    

    Is there statistical evidence to support an association between whether a freshman lives on campus and the amount of his or her school spirit?

Solutions

1. The response variable of interest is the customer's preferred sandwich choice. We may define

   p_T = proportion of customer's ordering turkey

   p_B = proportion of customer's ordering beef

   p_V = proportion of customer's ordering veggie

   p_S = proportion of customer's ordering salami

   p_H = proportion of customer's ordering ham

   p_P = proportion of customer's ordering pastrami

2. The set of hypotheses of interest is

   H₀: p_T = 0.28, p_B = 0.12, p_V = 0.20, p_S = 0.14, p_H = 0.22, p_P = 0.04

   H_a: not H₀

3. p- value = 0.0250
4. The p-value is between 0.05 and 0.1.
5. p-value < 0.0005
6. p-value > 0.3

7.  

   

8. 3.2452
9. p-value > 0.30
10. We would not reject the null hypothesis that the percentages of turkey, beef, veggie, salami, ham, and pastrami being ordered are as the sandwich shop owner has hypothesized.
11. H₀ : p_1j = p₁, p_2j = p₂, . . . p_rj, j = M, F where M stands for males and F stands for females.

    H_a : not H₀.

12.  

    

13. The conditions for the test are satisfied. The test statistic = 25.2554.
14. p-value < 0.0005
15. We reject the null hypothesis.
16. Evidence exists that males and females differ in the proportion of the types of vehicles they drive
17. X² = 36.9370; p < 0.001.We conclude to reject the null hypothesis. A relationship exists between whether or not a student lives on campus and how much school spirit he or she has.