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为什么不能拒绝H0？

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问题如下：

Question

The samples are drawn independently and both populations are assumed to be normally distributed

Using the above data, an analyst is trying to test the null hypothesis that the population variances are equal (H₀: $σ_{1}^{2} = σ_{2}^{2}$ ) against the alternative hypothesis that the variances are not equal (H_a: $σ_{1}^{2} \neq σ_{2}^{2}$ ) at the 5% level of significance. The table of the F-distribution is provided below.

Table

of the F-Distribution

Which of the following statements is most appropriate? The critical value is:

选项：

A.6.39 and reject the null.

B.9.60 and do not reject the null.

C.7.15 and do not reject the null.

解释：

Solution

B is correct. The test statistic makes use of the F-distribution and is the ratio of the variances, with the larger variance in the numerator. The test statistic is $F = s_{2}^{2} / s_{1}^{2}$ = 28/4 = 7. The degrees of freedom are 4 by 4. As it is a two-tailed test, the correct critical value at α = 5% is 9.60 (Panel B). As the test statistic is less than the critical value (i.e., 7 < 9.60), the null hypothesis cannot be rejected.

A is incorrect. It uses 4 by 4 degrees of freedom and does not divide the given alpha by two as is appropriate for a two-tailed test.

C is incorrect. It uses the 5 by 5 critical value (i.e., 5 degrees of freedom) rather than correctly using the 4 by 4 value (i.e., 4 degrees of freedom).

如果critical value小于H0, 为什么不能拒绝Null Hypothesis? 就因为test statistic小于critical value就应该不拒绝吗

为什么不能拒绝H0？

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