To test the claim that the variation in Friday afternoon taxi-cab ride times is greater than that of Friday morning times, we can perform an F-test for equality of variances.

Given:

• Friday Morning:
  • Population standard deviation ($\sigma$) = 10.4 minutes
• Friday Afternoon:
  • Sample size ($n$) = 13
  • Sample mean ($\overline{x}$) = 19.4 minutes
  • Sample standard deviation ($s$) = 12.7 minutes
• Significance level ($\alpha$) = 0.01

Hypotheses:

• Null Hypothesis ($H_0$): $\sigma^2_{\text{afternoon}} \leq \sigma^2_{\text{morning}}$
• Alternative Hypothesis ($H_a$): $\sigma^2_{\text{afternoon}} > \sigma^2_{\text{morning}}$

Test Statistic:

The test statistic for the F-test is calculated as:

$F = \frac{s^2_{\text{afternoon}}}{\sigma^2_{\text{morning}}}$

Calculating:

1. $s^2_{\text{afternoon}} = (12.7)^2 = 161.29$
2. $\sigma^2_{\text{morning}} = (10.4)^2 = 108.16$

Substituting into the F statistic formula:

$F = \frac{161.29}{108.16} \approx 1.49$

Critical Value:

Using an F-distribution table, we find the critical value for $\alpha = 0.01$ with degrees of freedom $df_1 = n - 1 = 12$ (afternoon sample) and $df_2$ which is based on the population standard deviation (in this case, it's assumed to be infinite).

From the F-distribution table:

• Critical value for $F_{0.01, 12, \infty}$ is approximately 2.67.

Conclusion:

• Compare $F$ to the critical value: $1.49 < 2.67$.
• Since the test statistic does not exceed the critical value, we fail to reject the null hypothesis.

Final Result:

At the 0.01 significance level, there is not enough evidence to conclude that the variation in Friday afternoon taxi-cab ride times is greater than that of Friday morning times.

Do you want more details on any part of this process? Here are some related questions you might consider:

1. What assumptions are made when performing an F-test?
2. How would the conclusion change if the significance level were higher?
3. What are the implications of failing to reject the null hypothesis in practical terms?
4. How can sample size affect the results of variance tests?
5. What are alternative tests for comparing variances?

Tip: Always check the underlying assumptions of the statistical test being used, as they can influence the validity of the results.