Chapter 17Chapter 17 Population means:Population means:

Two-Sample ProblemsTwo-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.

n s

Exercisers 29 66 8.6

Nonexercisers 31 75 9.0

This is an example of when to use the two-sample t procedures.

x

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).

Population

Variable

Mean Standard

deviation

Sample

Size

SampleMean

Sample

s.d.

1 x1 µ1 σ1 n1 s1

2 x2 µ2 σ2 n2 s2

1x

2x

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

the observed difference : 21 xx

2

22

1

21

nn

Problem: We don’t know the population standard deviations 1 and 2.

Solution: Estimate them with s1 and s2. The estimator is called the standard error.

2

22

1

21

n

s

n

sSE

The test statistic: Two-sample t statistic:

SE

xxt 21

Using p-values:

. value-p if HReject

1).-n 1,-min(n ison distributi t theof

freedom of degree theand level, cesignifican theis where

).t T P(2value-p : :H vs: (1)

); t T P(value-p : :H vs: (2)

); t T P(value-p : :H vs: (1)

0

21

obs21a210

obs21a210

obs21a210

H

H

H

Using critical valuesUsing critical values

1).-n 1,-min(n ison distributi t theof

freedom of degree theand level, cesignifican theis where

.T t ifHreject : :H vs: (1)

; tifHreject : :H vs: (2)

;T tifHreject : :H vs: (1)

21

2-1

obs 021a210

obs 021a210

-1obs 021a210

H

TH

H

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 118 126 126 120 129

16 weeks

124 98 110 140 110

population

Treatment

n s

1 2 weeks 5 123.80

4.60

2 16 weeks

5 116.40

16.09

x

21210 : vs.:

:hypothesisTest

aHH

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 n1 = n2 = 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