Chapter 17 Population means: Population means: Two-Sample Problems

Chapter Outline Two sample problems Comparing two population means Two-sample t procedures Examples of the two-sample t procedures Robustness of the t procedures

1. Two sample problems The units are not matched, and the samples can have different sizes.

When to use the two-sample test? A study is performed to compare the mean resting pulse rate of adult subjects who regularly exercise with the rate of those who do not regularly exercise. ns Exercisers Nonexercisers This is an example of when to use the two-sample t procedures.

Recall Matched pairs t procedures In a matched pairs design, subjects are matched in pairs and each treatment is given to one subject in each pair. Note: subjects paired; same sample size. Method: apply the one-sample t procedures to the observed difference.

Conditions for using two-sample t procedures

Notation: Want to test (  1 –  2 ). PopulationVariableMeanStandard deviation Sample Size Sample Mean Sample s.d. 1x1x1 µ1µ1 σ1σ1 n1n1 s1s1 2x2x2 µ2µ2 σ2σ2 n2n2 s2s2

2. Two-Sample t Procedures Definition: the standard deviation of the observed difference :

Problem: We don’t know the population standard deviations  1 and  2. Solution: Estimate them with s 1 and s 2. The estimator is called the standard error.

The test statistic: Two-sample t statistic:

Using p-values:

Using critical values

3. Examples Ex 17.3(P445) A researcher buried polyester strips in the soil for different lengths of time, then dug up the strips and measured the force required to break them. Here are the breaking strengths: 2 weeks weeks

populationTreatmentn s 12 weeks weeks

Example 17.4 (page 446)

4. Robustness of t Procedures The two-sample t procedures are more robust than the one-sample t methods, particularly when the distributions are not symmetric. When the two populations have similar distribution shapes, the probability values from the t table are quite accurate, even when the sample sizes are as small as n 1 = n 2 = 5.

Robustness: When the two populations have different distribution shapes, larger samples are needed. In planning a two-sample study, it is best to choose equal sample sizes. In this case, the probability values are most accurate.

More examples Ex 17.5, P444 Ex 17.23, P460 Ex 17.25, P461