Question

HOMEWORK #9: Two Sample Independent t-test Use the HW9 script. Get a random sample of n=20 from the data set. H0: ??−??=0 H1: ??−??≠0

Given:

Observed sample mean A= 13

Observed sample mean B= 16.6

Observed sample difference: -3.6

Pooled variance = 4.022222

Standard error of the difference of the sampling distribution= 0.8969082

Obtained t= -4.0138

Question:

1. Create a 95% confidence interval (I know the the answer is (-5.484334, -1.715666) but I don't know how to do it, so please step by step)

Answer 1

here sample size for population 1 is n1=20 and sample size for population 2 is n2=20

sample mean of first sample X1bar=13

sample mean of second sample X2bar=16.6

standard error of the difference of sampling distribution is s=0.8969082

so the t statistic for testing H0: ??−??=0 H1: ??−??≠0 is

T=((X1bar-X2bar)-(uA-uB))/s which under H0 follows a t distribution with df n1+n2-2

so P[|T|<t_{alpha/2,n1+n2-2}]=1-alpha since t distribution is a symmetric distribution. and t_{alpha/2.n1+n2-2} is the upper alpha/2 point of a t distribution with df n1+n2-2

so P[-t_{alpha/2,n1+n2-2}<T<t_{alpha/2,n1+n2-2}]=1-alpha

or, P[-t_{alpha/2,n1+n2-2}<((X1bar-X2bar)-(uA-uB))/s<t_{alpha/2,n1+n2-2}]=1-alpha

or, P[(X1bar-X2bar)-s*t_{alpha/2,n1+n2-2}<uA-uB<(X1bar-X2bar)+s*t_{alpha/2,n1+n2-2}]=1-alpha

so 100*(1-alpha)% confidence interval is [(X1bar-X2bar)-s*t_{alpha/2,n1+n2-2,}(X1bar-X2bar)+s*t_{alpha/2,n1+n2-2}]

so for 95% confidence interval alpha=0.05

now using R, t_{0.025,20+20-2}=2.024394

hence the required confidence interval is

[(13-16.6)-0.8969082*2.024394,(13-16.6)+0.8969082*2.024394]=[-5.4156955,-1.7843044] [answer]