Question

a restaurant manager suspects that service declines during off - peak hours. to investigate, he selects a random sample of 100 customers who dined in his restaurant during peak hours and a random sample of 70 customers who dined in his restaurant during off - peak hours. each customer rated the service on a scale of 1 to 5, where 1 = highly dissatisfied and 5 = highly satisfied. the results are displayed in the table.
the manager would like to test these hypotheses:
are the conditions for inference met?
no, the random condition is not met.
no, the 10% condition is not met.
no, the large counts condition is not met.
yes, all three conditions for inference are met.

Explanation:

• Random condition: The problem states that the manager selected a random sample of customers during peak hours and a random sample during off - peak hours. So, the random condition is met.
• 10% condition: In most cases (unless we know very specific information about the total number of customers in the restaurant), when sampling without replacement, if \(n_1 = 100\) (peak - hour sample) and \(n_2=70\) (off - peak hour sample), it is very likely that \(n_1\leq0.1N_1\) and \(n_2\leq0.1N_2\) (where \(N_1\) is the total number of peak - hour customers and \(N_2\) is the total number of off - peak hour customers). So, the 10% condition is met.
• Large Counts condition: For a chi - square test (which is appropriate for testing the distribution of categorical data like service ratings), the expected counts should be at least 5. The formula for the expected count \(E_{ij}=\frac{(R_i\times C_j)}{n}\), where \(R_i\) is the row total, \(C_j\) is the column total, and \(n\) is the grand total.
• The grand total \(n=100 + 70=170\).
• For example, for the cell (1, Peak) where the row total \(R_1=5 + 13=18\), column total \(C_1 = 5+8 + 12+40+35=100\), \(E_{11}=\frac{18\times100}{170}\approx10.59\gt5\).
• We can check other cells in a similar fashion. All expected counts will be greater than 5. So, the Large Counts condition is met.

Answer:

Yes, all three conditions for inference are met.