Answer:
Correct option is (a). 60.391 and 101.879; because the test statistic is in a critical region, the test rejects the null hypothesis.
Step-by-step explanation:
A Chi-square test for population variance is used to perform this test.
The standard deviation is, 2.0 minutes.
Then the variance is, 4.0 minutes.
The hypothesis for this test is:
H₀: The population variance of all commute times is equal to 4.0 minutes, i.e. σ² = 4.
Hₐ: The population variance of all commute times is not equal to 4.0 minutes, i.e. σ² ≠ 4.
The test statistic is:
[tex]\chi^{2}_{calc.}=\frac{(n-1)s^{2}}{\sigma^{2}}[/tex]
The critical region of this test is defined as:
Reject H₀ if [tex]\chi^{2}_{calc.}<\chi^{2}_{\alpha /2, (n-1)}[/tex] or [tex]\chi^{2}_{calc.}>\chi^{2}_{(1-\alpha /2), (n-1)}[/tex].
The degrees of freedom is:
[tex]n-1=81-1=80[/tex]
Compute the critical from a Chi-square table.
[tex]\chi^{2}_{\alpha /2, (n-1)}=\chi^{2}_{0.05, 80}=101.879\\\chi^{2}_{(1-\alpha /2), (n-1)}=\chi^{2}_{0.95, 80}=60.391\\[/tex]
The test statistic value is, [tex]\chi^{2}_{calc.}=105.8[/tex].
[tex]\chi^{2}_{calc.}=105.8 > \chi^{2}_{0.05, 80}=101.879[/tex]
The null hypothesis is rejected because the test statistic is in the critical region.
Thus, the correct option is (a).
