To test the claim that Friday afternoon times have greater variation than Friday morning times, we conduct a hypothesis test for the variance using an $F$-test. Here are the steps:

Step 1: Define the Hypotheses

Let:

• $\sigma^2$ represent the variance of Friday afternoon times,
• $\sigma_0^2 = 10.2^2 = 104.04$ represent the variance of Friday morning times.

The hypotheses are:

• Null Hypothesis $H_0$: $\sigma^2 = \sigma_0^2$ (Friday afternoon times have the same or lower variance than Friday morning times),
• Alternative Hypothesis $H_1$: $\sigma^2 > \sigma_0^2$ (Friday afternoon times have greater variance than Friday morning times).

Step 2: Set the Significance Level

The significance level $\alpha = 0.05$.

Step 3: Compute the Test Statistic

We use the formula for the test statistic $F$: $F = \frac{s^2}{\sigma_0^2}$ where:

• $s = 13.9$ is the sample standard deviation of Friday afternoon times,
• $\sigma_0 = 10.2$.

Calculating $s^2$ and then the $F$-statistic: $s^2 = 13.9^2 = 193.21$ $F = \frac{193.21}{104.04} \approx 1.857$

Step 4: Determine the Critical Value

Since we are conducting a one-tailed test, we find the critical $F$-value for $\alpha = 0.05$, with degrees of freedom:

• $df_{\text{numerator}} = n - 1 = 11 - 1 = 10$,
• $df_{\text{denominator}} = \infty$ (or large-sample approximation for Friday morning times).

Using an $F$-distribution table or calculator, we find that the critical value for $F$ at $df = 10$ and a one-tailed $\alpha = 0.05$ is approximately $F_{\text{critical}} = 2.978$.

Step 5: Make a Decision

• If $F \geq F_{\text{critical}}$, reject $H_0$.
• If $F < F_{\text{critical}}$, fail to reject $H_0$.

Since $1.857 < 2.978$, we fail to reject $H_0$.

Conclusion

At a 0.05 significance level, there is insufficient evidence to support the claim that Friday afternoon taxi-cab ride times have greater variation than Friday morning times.

Would you like further details on any steps or clarification?

Here are five related questions to extend your understanding:

1. How would the result change if the sample size were larger?
2. What would happen if the significance level was changed to 0.01?
3. How do we interpret the $F$-distribution in terms of variance?
4. What assumptions are critical for the validity of this $F$-test?
5. How would this test differ if we wanted to test for a smaller variance instead?

Tip: The $F$-test for variance is sensitive to normality. Small deviations from a normal distribution can affect the test outcome.