5. Sample Size Calculation: Consider a random sample of size $n$ from a normal distribution $X \sim N(\mu, 4)$. 1 (a) Test $H_0: \mu = 18$ versus $H_a: \mu > 18$ at the significance level $\alpha = 0.05$. (b) What is $\beta(\mu)$ the probability of Type II error at $\mu$. Compute the probability of Type II error when $\mu = 20$. Does it depend on $n$? What is $\beta(20)$ when $n = 9$? What is $\beta(20)$ when $n = 16$? What is the smallest value of $n$ such that $\beta(20) = .01$?
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In this case, the null hypothesis is p = 18 and the alternative hypothesis is p > 18. The probability of Type II error depends on the true value of p, the sample size n, and the significance level α. We can use the formula for the power of the test: power = 1 - β Show more…
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