= Solution
\b[Under the question's stipulated nonzero population difference, failing to reject is a <type II error>.] A <type I error> would instead reject a true equal-mean <null hypothesis>. The premise concerns the underlying difference: another study's <statistical significance>, by itself, would not prove that its conclusion is correct.
For a specified alternative and sampling design, write
$$
\beta=\mathbb P_{\rm alternative}(\text{do not reject }H_0),\qquad
1-\beta=\mathbb P_{\rm alternative}(\text{reject }H_0).
$$
The second probability is <statistical power>. Across repeated <random samples> under that same alternative and design, the long-run frequency of this kind of error is $\beta$.
\b[No numerical value of $\beta$ follows from this output.] It depends on the true <effect size>, group <variances>, <sample sizes>, decision rule and chosen <significance level>. In particular, it is neither $0.9561$, nor $1-0.9561$, nor automatically 5%. The reusable result that <a p-value does not determine the type II error probability> demonstrates this directly in a simple normal testing model.
A small <effect size> or an imprecise estimator can produce low <statistical power>. The present <confidence interval> describes the precision of the estimated difference; it does not supply the missing true alternative needed to calculate the <power function>. A specified 5% <significance level> concerns the <type I error> rate under the null, rather than a universal error rate whenever a result is nonsignificant.
Back to article page
