Suppose that a population follows a normal distribution with unknown mean u and

Question

Suppose that a population follows a normal distribution with unknown mean u and known standard deviation = 2, and we want to test the null hypothesis that H0: u=10 versus the alternative that Ha : u > 10 at significant level .05.

a.) Based on a sample size of 25, for what values of x will H0 be rejected? (That is, what is the rejection region for this test?)

b.) Suppose that the true value of the population mean is u = 10 and we collect a random sample of size 25 from the population. What is the probability that we will reject H0 if we perform a significance test of H0 : u = 10 verses Ha : u > 10?

c.) Suppose that the true value of the population mean is u = 11 and we collect a random sample of size 25 from the population. What is the power of the significance test of H0: u=10 versus Ha: u > 10 under this alternative? (That is, what is the probablity that we will reject H0 under this alternative?)

Explanation / Answer

(A)
for 0.05 significance level, z-value = 1.64
Hence the threshold value of x beyond which null hypothesis be rejected is
x = mean + z*SE
SE = sigma/sqrt(n)
SE = 2/sqrt(5)
SE = 0.8944
x = 10 + 1.64*0.8944 = 11.4668
Hence for the values of x greater than 11.4668, null hypothesis be rejected

(B)
For a hypothesis test, probability that the null hypothesis will be rejected is given by the significance level, alpha. In this hypothesis test 0.05 is the significance level. Hence probability that null hypothesis, H0 will be rejected is 0.05.

(C)
For mu = 11, calcualte the probability that x < 11.4668
P(x < 11.4668) = P(z < (11.4668 - 11)/0.8944) = P(z<0.5219) = 0.6991
Hence power of test = 1 - 0.6991 = 0.3009

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