4. A scientist is doing a preliminary study to try to determine the sample size necessary for her larger

study. She would like to show that the mean in her population of interest is larger than 12 and is

starting with the assumption that Ho : = 12which is will be testing at the = 0.1level. She takes a

random sample of n=10 from her population and checks the QQplot and sample histogram. From the

graphs, the normality of the population assumption is pretty well met. Her sample mean is X bar = 14.2

and sample standard deviation is s = 4.88 .

(a) Compute the critical values and rejection region for the appropriate test statistic and X barand p

value of the appropriate test. Draw a conclusion in the context of the study.

(b) Suppose she is interested in the specific alternative that A = 15and will still be conducting a

one-sided test. Compute the power of a follow up hypothesis test to correctly reject the null if in

fact A = 15is true with a sample size of 40. Use the sample standard deviation above to estimate

, use zcritical values, and = 0.1 .

(c) She will use s = 4.88as her best guess of so she can use a Z test simplification. Approximately

what sample size would be required to achieve a power of 0.85 if the true population mean is

A = 15 ? Give your answer as the smallest whole number that meets the criterion.

(d) If she decided to perform the test at a significance level of = 0.05instead of = 0.10 , how would

that effect her power of detecting A = 15 ? (no calculations needed)