6.3 Two-Sample Inference for Means

November 17, 2003

Paired Differences

Matched Pairs Designexperimental plan where the

experimental units are divided into halves and two treatments are randomly assigned to the halves

Attempting to determine if there is a significant difference between the mean responses of the treatments

Procedure

Obtain the difference between responses for each experimental unit

Analyze the differences using a one-sample approachIf a large sample is obtained, use

critical values from the Standard Normal distribution (z)

Otherwise, use critical values from the corresponding t distribution

Example 9 pg 370

Students worked with a company on the monitoring of the operation of an end-cut router in the manufacture of a wood product. They measured the critical dimensions of a number of pieces of a type as they came off the router. Both a leading-edge and a trailing-edge measurement were made on each piece. Both were to have a target value of .172 in.

Piece Leading Edge

Trailing Edge

1 .168 .169

2 .170 .168

3 .165 .168

4 .165 .168

5 .170 .168

Piece Leading Edge

Trailing Edge

Difference

1 .168 .169 -.001

2 .170 .168 .002

3 .165 .168 -.003

4 .165 .168 -.003

5 .170 .168 .002

Confidence Interval

n

stdor

n

szd dd

0023.0008. dsd

Hypothesis Test

Is there a significant difference between the measurements at α =.01?

Ho: µ = 0 Ha: µ ≠ 0

n

sd

dT

#

Independent Samples

The goal of this type of inference is to compare the mean response of two variables (or treatments) when the data are not paired or matched

A key assumption that will be made is that the separate samples used to collect information concerning the two variables are independent One sample does not influence the other

sample in any way Furthermore, one must assume that both

sampled populations are normally distributed

Large Sample Comparison

The quantity of interest is a linear combination of population means, namely µ1 - µ2

The above quantity will be estimated by As a result, various quantities of the sampling

distribution of the difference, under the assumption of equality between the means, need to be developed

Ho: µ1 - µ2 = #

Ha: µ1 - µ2 ≠ #

21 xx

Test Statistic

2

22

1

21

#)( 21

ns

ns

xxZ

Confidence Interval

2

22

1

21)( 21 n

snsZxx

Example A company research effort involved finding

a workable geometry for molded pieces of a solid. A comparison was made between the weight of molded pieces and the weight of irregularly shaped pieces that could be poured into the same container. A series of 30 attempts to pack both the molded and the irregular pieces of the solid were compared. Is there enough evidence to suggest that the irregular pieces produced higher weights?

The Data

1 = molded n1 = 30

s1= 9.31 = 164.65

2=irregular n2 = 30

s2= 8.51 = 179.651x 2x

Small Samples

If at least one sample size is small, then use critical values from a t distribution for constructing confidence intervals and performing hypothesis tests with degrees of freedom obtained by Satterthwaite’s Approximation

Satterthwaite’s Approximation

2

2

22

2

2

1

21

1

2

2

22

1

21

11

11

ˆ

ns

nns

n

ns

ns

Test Statistic

21

11

21 #)(

nnPs

xxT

2

)1()1(

21

222

2112

nn

snsnsP

Confidence Interval

21

1121 )( nnPstxx

Example

The data shown gives spring lifetimes under two different levels of stress (900 and 950 N/mm2). Do the data give evidence of a significant difference at α = .05?

950 Level 900 Level

225 171 198 187 189 135 162 135 117 162

216 162 153 216 225 216 306 225 243 189

Minitab Output

Descriptive Statistics: 950, 900

Variable N Mean Median TrMean StDev SE Mean950 10 168.3 166.5 167.6 33.1 10.5900 10 215.1 216.0 211.5 42.9 13.6

Assignment

Page 385: #3, #4