 Solution

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, writeThe 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 $\beta$.

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.