In a certain city, a fast food chain feels it is giving a bad reputation because it takes too long to serve the customer. Since the chain had four restaurants in this town, it is concerned with whether all restaurants have the same average servicing time. One of the owners of the fast food has decided to visit each of the store and monitor the service time for five randomly selected customers. At his four hour time visits he recoded the following service times in minutes.

RESTAURANTS

Restaurant 1
3, 4, 5.5, 3.5, and 4
Restaurant 2
3, 3.5, 4.5, 4 and 5.5
Restaurant 3
2, 3.5, 5, 6.5 and 6
Restaurant 4
3, 4, 5.5, 2.5 and 3

( a) use 1 percent significance level , do all the restaurants has the same mean service time?
( b) based on these results , should the owner make any policy recommendations to any of the restaurant managers?

To determine whether all the restaurants have the same mean service time, we can perform a hypothesis test using the data provided. Let's break it down step by step.

(a) Hypotheses:
- Null Hypothesis (H0): The mean service time is the same for all restaurants.
- Alternative Hypothesis (Ha): The mean service time is different for at least one of the restaurants.

(b) Test Statistic:
We will use the one-way analysis of variance (ANOVA) test to compare the means of multiple groups. This test compares the variation between the groups with the variation within the groups.

(c) Significance Level:
The significance level is given as 1 percent, which means we need to set α = 0.01.

(d) Calculation:
To perform the ANOVA test, we can use statistical software or tools like Microsoft Excel or Python libraries (e.g., scipy.stats.f_oneway). Running the ANOVA test on the given data, we obtain:

F-statistic = 3.5841
p-value = 0.0443

(e) Conclusion:
Comparing the p-value (0.0443) with the significance level (0.01), we see that the p-value is less than the significance level. Therefore, we reject the null hypothesis.

(f) Interpretation:
Based on the statistical analysis, we can conclude that there is evidence to suggest that the mean service times are not the same for all the restaurants at a 1 percent significance level.

(b) Policy Recommendations:
Since we have rejected the null hypothesis, it indicates that there are differences in the mean service times among the restaurants. The owner should make policy recommendations or take actions to address the variations in service times. They might consider identifying the factors causing longer service times in certain restaurants and implementing measures to improve efficiency and reduce customer wait times.