11.2 COMPARING TWO MEANS (Pages 617 -633) OVERVIEW: One can use thet-distribution on two samples that are not matched pairs. (Hencethere would be no difference column.) Basically, one tests the nullhypothesis that the two samples came from populations with equalmeans. The degrees of freedom is the smaller of n
1 -1, n2 -1, where n1 and n2 are the sample sizes.Let
x
1 (bar) = mean of sample ofsize n1 frompopulation P1s1
= standard deviation calculated from the sample x2
(bar) = mean of sample of size n 2 from populationP2s2
= standard deviation calculated from the sample m
1 = unknown mean of populationP1m
2 = unknown mean of populationP2To test the null hypothesis m1
= m 2 , we calculate the tstatistict = [x
1 (bar)-x2 (bar)]/[sqrt(s 1 2/n1 + s2 2 /n2 )],with df = the smaller of n
1 -1, n2 -1.The reader is reminded that, in this situation,"variances add" (see page 400). The expression under the radical inthe denominator is an approximation for the variance of the samplingdistribution of mean differences. Hence, the square root is anapproximation for the standard deviation of the meandifferences.
A confidence interval for the difference ofpopulation means has the form
(x
1 (bar)-x2 (bar)) plus/minust*[sqrt(s12 /n1 + s2 2/n2 )]. Example:
Here are hypothetical results of a computerprogramming aptitude test that was administered to random samples of51 men and 60 women.
Men
nm = 51
x(bar)m = 82
sm = 5
Women
nw = 60
x(bar)w = 84
sw = 7
Null hypothesis H
0 : mm= m w Alternate hypothesis H
a : mmis not equal to mw Degrees of freedom = 51-1 = 50.
t = (82-84)/sqrt(25/51 + 49/60) = -1.75
If we test Ho at the 5% level of significance(2-tail), the critical region is t < -2.009 or t > 2.009. Sincethe calculated t is not in the critical region, the difference 82-84is not significant at this level.
We can also note that
tcdf( -1E99,-1.75,50) =.0431266224, which also indicates that we are not in the "outer 2.5%"in a 2-tail test.We could construct a 95% confidence interval form
m - mw. This interval is
(82-84) plus/minus (2.009)sqrt(25/51 + 49/60) = -2 plus/minus 2.30.This is the interval [-4.30, 0.30].
Note that this interval contains 0, supporting thepreviously obtained conclusion that the mean difference of -2 is notsignificant at the 5% level.