INFERENCES FOR CATEGORICAL DATA
In this lab assignment, you will apply the summary and inferential tools for categorical data in Excel to data from a restaurant in Edmonton. In particular, you will obtain summary statistics to describe the data and confidence intervals for the proportion of customers who would recommend this restaurant. You will also make inferences about the difference in the proportion of customers who would recommend this restaurant during peak hours or off-peak hours. In addition, we will observe which section of the menu the customers are most interested in. Finally, the relationship between the customers’ preference of the menu section and their recommendation of the restaurant is examined. The assumptions required to make the inferences valid will be discussed as well.
Restaurant
The local restaurant owner wants to learn about the satisfaction of his customers on his lunch menu. His restaurant usually serves over 50 customers each day during lunch. He decided to ask all his servers to include a survey card in the check folder at the end of their lunch meal over 3 months. This survey card includes only 3 questions:
a) Did you visit during peak hours (noon to 1pm)?
b) Which section of the menu (Appetizer/Main Dish/Dessert) do you like the most?
c) Will you recommend this restaurant to others?
By the end of the survey period, the owner collected 150 responses. The data are available in the Excel file
lab3.xlsx. The data are not to be printed in your submission The following is a description of the variables in the data file: Variable Description of Variable
Peak Hour Yes or No,
Menu Preference of the menu (Appetizer/Main Dish/Dessert) Response Recommend or Not recommend the restaurant to others.
1. First, you will discuss the survey design.
(a) What are possible sources of bias in the survey? Explain briefly.
(b) Can we generalize the results of the survey to all customers in this restaurant?
2. Now you will display and summarize the data with a bar chart and frequency table.
Tips: Use the PivotTable feature to obtain answers for this question.
(a) Obtain a relative frequency bar chart of response (recommend or not recommend) for the 150 survey. Paste the plot into your report.
(b) Obtain a frequency table of response (showing both frequency and relative frequency). Paste the table into your report. What was the sample proportion of customers that will recommend the restaurant to others? What was the sample proportion of customers that will not recommend the restaurant to others?
3. Now you will estimate the proportion of customers that will recommend the restaurant to others with 95% confidence interval.
Tips: There is no function in Excel that can do it for you, so you need to calculate all values by yourself.
(a) Obtain and interpret a 95% confidence interval for the proportion of customers that will recommend the restaurant to others.
(b) What are the assumptions required to make the inferences in part (a) valid? Are the assumptions met in this case? Explain briefly.
4. Now you will examine the change in the proportion of customers that will recommend the restaurant to others between peak hours and non-peak hours.
(a) Is there evidence that the restaurant owner has seen a change in the proportion of customers that will recommend the restaurant to others between peak hours and non-peak hours? Obtain a contingency table (using the PivotTable feature) of the count of recommendations and not recommendations by peak hours. Paste the table into your report.
(b) Refer to the output in part (a) to answer the following questions: What percent of customers that would recommend the restaurant came during peak hours? What percent of customers that would recommend the restaurant didn’t come during peak hours?
(c) Carry out the appropriate two-sample proportion test at the 5% significance level to answer the question. Paste the corresponding output into your report. In particular, state the null and alternative hypotheses, report the value of the test statistic, its distribution under the null hypothesis, and the p-value of the test. State your conclusions.
(d) Obtain and interpret a 95% confidence interval for the difference in the proportion of customers that will recommend the restaurant to others between peak hours and non-peak hours. Paste the output into your report. What is the estimated difference in the proportions? Is the interval consistent with the conclusion reached in part (a)?
(e) Are the assumptions for inference in parts (a) and (b) met in this case? Refer to theory to answer the question.
(f) Is there evidence that the restaurant owner has seen a change in the proportion of customers that will recommend the restaurant to others between peak hours and non-peak hours? Carry out the appropriate chi-square test for homogeneity at the 5% significance level to answer the question. Paste the corresponding output into your report. In particular, state the null and alternative hypotheses, report the value of the test statistic, its distribution under the null hypothesis, and the p-value of the test. State your conclusions.
(g) What is the relationship between the tests in parts (a) and (d)? Particularly, comment on the relationship between their test statistic values, p-values, and conclusions.
5. Now you will examine the relationship between customers’ preference section of the menu and their recommendation of the restaurant. Using 𝛼 = 0.01, carry out an appropriate test to test if there is a relationship between customers’ preference section of the menu and their recommendation of the restaurant. State the null and alternative hypotheses, report the value of the test statistic, its distribution under the null hypothesis, and the p-value of the test. State your conclusions.
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