"A knowledge of statistics is like aknowledge of foreign languages or of algebra; it may prove of use atany time under any circumstances."

Arthur L. Bowles, Elements of Statistics

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 n1-1, n2-1, where n1 and n2 are the sample sizes.

Let

x1(bar) = mean of sample ofsize n1 frompopulation P1

s1 = standard deviation calculated from the sample

x2(bar) = mean of sample of size n2 from populationP2

s2 = standard deviation calculated from the sample

m1 = unknown mean of populationP1

m2 = unknown mean of populationP2

To test the null hypothesis m1 = m2, we calculate the tstatistic

t = [x1(bar)-x2(bar)]/[sqrt(s12/n1 + s22/n2)],

with df = the smaller of n1-1, 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

(x1(bar)-x2(bar)) plus/minust*[sqrt(s12/n1 + s22/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 H0: mm = mw

Alternate hypothesis Ha: mm is 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 formm - 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.