In a sample of 25 adolescents in an immunologic study, one variable of interest was the diameter of skin test reaction to an antigen. The sample mean and standard deviation were 24 and 15 mm, respectively.
a) Conduct an appropriate test (state H0, HA and critical value) to see if we can conclude that the population mean of the diameter of skin test reaction to this antigen is different from 30, use α=0.05. Report the p-value of this test.
b) Construct a 95% confidence interval for the population mean (μ) of the diameter of skin test reaction to this antigen in this population. Based on the 95% confidence interval, would
you have expected to reject or not to reject the null hypothesis in part (a)? Why?
c) In another sample of size 100, the 95% confidence interval of diameter for skin test reaction is found to be (18.25, 25.75). What are the sample mean and sample standard deviation of the diameter for skin test reaction to this antigen of this sample?
d) The true population mean for the diameter of skin test reaction to this antigen is 25 mm, and the population variance for this variable is 100. If you want to have a 90% power to detect this difference, using a two-sided test conducted at the 0.01 level of significance, what sample size will be needed?