To determine the critical values and make a conclusion about the hypothesis test regarding the standard deviation of commute times, we start by identifying the appropriate test and critical values given the significance level.
Hypothesis Test
We are testing the following hypotheses:
- Null Hypothesis \( H_0: \sigma = 2.0 \) (the population standard deviation is equal to 2.0 minutes)
- Alternative Hypothesis \( H_a: \sigma \neq 2.0 \) (the population standard deviation is not equal to 2.0 minutes)
Sample Statistics
- Sample size \( n = 81 \)
- Sample standard deviation \( s = 2.3 \)
- Test statistic \( \chi^2 = 105.8 \)
- Significance level \( \alpha = 0.10 \)
Degrees of Freedom
The degrees of freedom for the chi-square test is:
\[
df = n - 1 = 81 - 1 = 80
\]
Critical Values
For a two-tailed test with \( \alpha = 0.10 \), we split the significance level into two tails: each tail has \( \alpha/2 = 0.05 \).
We need to find the critical chi-square values for \( \alpha = 0.05 \) in both tails with 80 degrees of freedom.
Using a chi-square table or calculator:
- The critical value for \( \chi^2_{0.05, 80} \) (the upper critical value) is approximately \( 113.145 \).
- The critical value for \( \chi^2_{0.95, 80} \) (the lower critical value) is approximately \( 69.126 \).
Conclusion
The critical values are:
- Lower Critical Value: 69.126
- Upper Critical Value: 113.145
Now, we check the test statistic:
- Test Statistic: \( 105.8 \)
Since \( 69.126 < 105.8 < 113.145 \), the test statistic falls within the acceptance region (i.e., it is not in the critical region).
Final Statement
Thus, because the test statistic is outside of the critical region, we fail to reject the null hypothesis.
Correct Answer:
B) 69.126 and 113.145; because the test statistic is outside of the critical region, the test fails to reject the null hypothesis.