Question

One aspect of queuing theory is to consider waiting time in lines. A fast-food chain is trying to determine whether it should switch from having four cash registers with four separate lines to four cash registers with a single line. It has been determined that the mean wait time in both lines is equal. However, the chain is uncertain about which line has less variability in wait time. From experience, the chain knows that the wait times in four separate lines are normally distributed with α = 1.2 minutes. In a study, the chain reconfigured five restaurants to have a single line and measured the wait times for 50 randomly selected customers. The sample standard deviation was determined to be s= 0.84 minutes. Is the variability in wait time less for a single line than for multiple lines at the α of 0.05?

What are we looking for and what are we going to do with this?

Answer 1

The question is about using hypothesis testing to determine if a single queue results in significantly less variability in wait times compared to multiple queues in a fast-food chain setup. An F-test is conducted to compare the sample standard deviation with the known population standard deviation at a significance level of α = 0.05.

Step-by-step explanation:

We are looking to determine if the variability, specifically the standard deviation, of customer wait times in a single queue is significantly less than the variability in multiple queues. We will utilize hypothesis testing to compare the sample standard deviation from the single queue with the population standard deviation from multiple queues.

To conduct this test, we set up null and alternative hypotheses as follows:

• H0: σ² ≥ σ²0 (The variance in a single line is greater than or equal to the variance with multiple lines)
• Ha: σ² < σ²0 (The variance in a single line is less than the variance with multiple lines)

We use the sample standard deviation (s) as our random variable. Since we are given a sample standard deviation (s = 0.84 minutes) and a population standard deviation (σ = 1.2 minutes) for the multiple lines, we can perform a one-tailed F-test to see if the difference in variability is statistically significant at the α = 0.05 level.

If the calculated F-value is less than the critical F-value from the F-distribution table, we can reject the null hypothesis and conclude that there is sufficient evidence to claim that the customer waiting times vary less with a single line compared to multiple lines.