AP Stats - Ch. 24

Comparing Means

Two sample t methods

Two sample t methods allow us to draw conclusions about the difference between the means of two independent groups. The two sample methods make relatively few assumptions about the underlying population, so they are usually the method of choice for comparing to sample means. However, the student’s t-models are only approximations for their true sampling distribution. To make the approximation work well, the two sample t methods have a special rule for estimating degrees of freedom.

two-sample t-interval for the difference between two means

a confidence interval for the difference between the mans of two independent groups found as ((bar)y1 - (bar) y2) +/- SE( (bar) y1 - (bar) y2)

where

SE ((bar) y1 - (bar) y2) = (root) s1^2/n1 + s2^2

and the number of degrees of freedom is given by a special formula

Two-sample t-test for the difference between means

the hypothesis test for the difference between the mean of two independent groups. It test the null hypothesis

\
H0= mue1 - mue2 =0 or mue1= mue2

where the hypothesized difference, delta 0, is almost always 0, using the statistic

tdf=(((bar)u1 - (bar) y2) - delta 0)/SE((bar) y1 - (bar) y2)

with the number of degrees of freedom given by a special formula

Pooling

Data from two or more populations may sometimes be combines, or pooled, to estimate a statistic (typically a pooled variance) when we are willing to assume that the estimated value is the same in both populations. THis resulting larger sample size may lead to an estimate with lower sampling variance. However, pooled estimates are only ok when the required assumptions are true.

Pooled t-methods

Pooles-t methods provide inferences about the difference between the means of two independent populations under the assumption that both populations have the same standard deviation. When the assumption is justified, pooled-t methods generally produce slightly narrower confidence intervals and more powerful significance tests than two-sample t methods. When the assumption is not justified, they generally produce worse results, sometimes substantially worse.

\
it is better to just use a two-sample t method instead