Question: A sample of size \(n=40\) is taken from a \(N\left(\mu, 10^{2}\right)\) population, and the data analysed by a consultant who, without taking statistical advice, has decided to reject \(H_{0}: \mu=10\) if \(\bar{X}<8\) or \(\bar{X}>12 .\) What is the probability that this rule results in the consultant making a Type I error?

Having taking some statistical advice, the consultant decides to change the rule to one where the null hypothesis will be rejected if \(\bar{X}<10-c\) or \(\bar{X}>10+c\). What value should \(c\) take to make the probability of a Type I error equal to $0.05$ ?

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