Question

As a researcher, your goal is to estimate the effect of a drug on test scores of human subjects undertaking a test in Statistics. All subjects were divided into two populations X and Y , with members of population X receiving the drug prior to testing and members of population Y receiving a placebo prior to testing. 16 subjects were chosen randomly from population X and the mean score in this group was equal to 9.78, with the sample standard deviation being 3.24. Another group of 9 subjects, chosen randomly from population Y , was used as a control group and had a mean score of 15.1, with the sample standard deviation of 4.17. Assume that test scores in both populations are distributed as normal with X ∼ N (µX, σ2 X) and Y ∼ N (µY , σ2 Y ), where µX, µY are unknown population means and σ 2 X, σ 2 Y are unknown population variances. Note: you will need to use computer software (Excel, Stata) or scientific calculators to obtain exact answers to some of the questions below. 1. Suppose your colleague has provided you with the values for the true population variances: σ 2 X = 9 and σ 2 Y = 16. Using this information, calculate 3

(a) 95% confidence interval for µX

(b) 95% confidence interval for µY

(c) p-value of a hypothesis test of H0 : µX = 8.5 vs Ha : µX > 8.5

(d) p-value of a hypothesis test of H0 : µY = 17.5 vs Ha : µY 6= 17.5

Answer 1

when we know population variance

we can use 1-sample z-test to calculate confidence interval

a)

n = 16 , sample mean= 9.78 , sd = 3

95% confidence interval

(8.310, 11.250)

b)

The assumed standard deviation = 4

N   Mean SE Mean      95% CI
9 15.10     1.33 (12.49, 17.71)

c)

TS = 1.71

p-value = 0.044

d)

TS = -1.80

p-value = 0.036