Question: An initial sample of size 25 from a Normal population with mean \(\mu\) and standard deviation \(\sigma=10\) gives \(\bar{X}=54.4 .\) Show that the test-statistic for testing the null hypothesis \(H_{0}: \mu=50\) against the two-sided alternative hypothesis \(H_{1}: \mu \neq 50\) leads to a p-value less than $0.05$.

A further sample of size 75 is then taken from the same population, and it is proposed to reconsider the same hypothesis using the combined sample of 100 values. What values of the mean of these 100 values will lead to p-value greater than $0.05$ when testing \(H_{0}\) for the combined sample?

Consider the further sample: determine the values of the sample mean in this further sample which will make the final p-value greater than $0.05$, and find the probability of obtaining a sample mean for the further sample in this range.

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