Chi-square test for population variance:

• Given $n = 20$, sample variance $s^2 = 25$, and hypothesized population standard deviation $\sigma_0 = 8 \implies \sigma_0^2 = 64$. The chi-square test statistic is $\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2} = \frac{(20 - 1) \times 25}{64} = \frac{19 \times 25}{64} = \frac{475}{64} \approx 7.4219 \approx 7.42$.
• $15.62$ results from using incorrect degrees of freedom.
• $51.20$ is an erroneous computation.
• $55.60$ is an incorrect distractor.

Relationship between significance levels in hypothesis testing:

• Rejection depends on whether the p-value is smaller than $\alpha$.
• If $\alpha = 0.025$ led to rejection, the p-value is $\le 0.025$. However, if $p = 0.018$, it is rejected at $0.025$ but not at $0.01$.
• If $p = 0.005$, it is rejected at both $0.025$ and $0.01$.
• Since the p-value is only known to be $\le 0.025$, it could be either less than $0.01$ or greater than $0.01$. Hence, the hypothesis may or may not be rejected at the 0.01 level.