Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Statistical Inferences Based on Two Samples Chapter 10.

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

McGraw-Hill/Irwin

Statistical Inferences Based on

Two Samples

Chapter 10

10-2

Chapter Outline

10.1 Comparing Two Population Means by Using Independent Samples: Variances Known

10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown

10.3 Paired Difference Experiments10.4 Comparing Two Population Proportions

by Using Large, Independent Samples10.5 Comparing Two Population Variances

by Using Independent Samples

10-3

10.1 Comparing Two Population Means by Using Independent Samples: Variances Known

Suppose a random sample has been taken from each of two different populations

Suppose that the populations are independent of each other Then the random samples are independent of

each other

Then the sampling distribution of the difference in sample means is normally distributed

10-4

Sampling Distribution of theDifference of Two Sample Means #1

Suppose population 1 has mean µ1 and variance σ1

2

From population 1, a random sample of size n1 is selected which has mean 1 and variance s1

2

Suppose population 2 has mean µ2 and variance σ2

2

From population 2, a random sample of size n2 is selected which has mean 2 and variance s2

2

Then the sample distribution of the difference of two sample means…

10-5


Is normal, if each of the sampled populations is normalApproximately normal if the sample

sizes n1 and n2 are large

Has mean µ1–2 = µ1 – µ2

Has standard deviation

2

22

1

21

21 nnxx

10-6


Figure 10.1

10-7

z-Based Confidence Interval for the Difference in Means (Variances Known)

A 100(1 – ) percent confidence interval for the difference in populations µ1–µ2 is

2

22

1

21

221 nnzxx

10-8

z-Based Test About the Difference in Means (Variances Known)

Test the null hypothesis aboutH0: µ1 – µ2 = D0

D0 = µ1 – µ2 is the claimed difference between the population means

D0 is a number whose value varies depending on the situation

Often D0 = 0, and the null means that there is no difference between the population means

10-9


Use the notation from the confidence interval statement on a prior slide

Assume that each sampled population is normal or that the samples sizes n1 and n2 are large

10-10

Test Statistic (Variances Known)

The test statistic is

The sampling distribution of this statistic is a standard normal distribution

If the populations are normal and the samples are independent ...

2

22

1

21

021

nn

Dxxz

10-11


Reject H0: µ1 – µ2 = D0 in favor of a particular alternative hypothesis at a level of significance if the appropriate rejection point rule holds or if the corresponding p-value is less than

Rules are on the next slide…

10-12

z-Based Test About the Difference in Means (Variances Known) Continued

10-13

Example 10.2: The Bank Customer Waiting Time Case

21.14

1009.1

1007.4

014.579.8

0:

0:

2

22

1

21

021

21

210

nn

Dxxz

H

H

a

10-14

10.2 Comparing Two Population Means by Using Independent Samples: Variances Unknown

Generally, the true values of the population variances σ1

2 and σ22

are not knownThey have to be estimated from

the sample variances s12 and s2

2, respectively

10-15

Comparing Two Population Means Continued

Also need to estimate the standard deviation of the sampling distribution of the difference between sample means

Two approaches:1. If it can be assumed that σ1

2 = σ22 = σ2,

then calculate the “pooled estimate” of σ2

2. If σ12 ≠ σ2

2, then use approximate methods

10-16

Pooled Estimate of σ2

21

2

21

222

2112

11

2

11

21 nns

nn

snsns

pxx

p

10-17

t-Based Confidence Interval for the Difference in Means (Variances Unknown)

2

11

11

21

222

2112

21

2221

nn

snsns

nnstxx

p

p

10-18

Example 10.3: The Catalyst Comparison Case

22.91,38.3042.302.750811

42.305

1

5

11.435

11

1.435255

2.4841538615

2

11

21

21

21

21

2

21

222

2112

xx

pxx

p

xx

nns

nn

snsns

10-19

t-Based Test About the Difference in Means: Variances Equal

21

2

021

11

nns

Dxx

p

10-20


6087.4

51

51

1.435

02.750811

11

0:

0:

21

2

021

211

210

nns

Dxxt

H

H

p

10-21

t-Based Confidence Intervals and Tests for Differences with Unequal Variances

11 2

2

222

1

2

121

2

2221

21

2

22

1

21

021

2

22

1

21

/221

n/ns

n/ns

/ns/nsdf

ns

ns

Dxxt

n

s

n

stxx

10-22

10.3 Paired Difference Experiments

Before, drew random samples from two different populations

Now, have two different processes (or methods)

Draw one random sample of units and use those units to obtain the results of each process

10-23

Paired Difference Experiments Continued

For instance, use the same individuals for the results from one process vs. the results from the other processE.g., use the same individuals to compare

“before” and “after” treatmentsUsing the same individuals, eliminates

any differences in the individuals themselves and just comparing the results from the two processes

10-24

Paired Difference Experiments #3

Let µd be the mean of population of paired differences µd = µ1 – µ2, where µ1 is the mean of population 1

and µ2 is the mean of population 2

Let d̄ and sd be the mean and standard deviation of a sample of paired differences that has been randomly selected from the population d̄ is the mean of the differences between pairs of

values from both samples

10-25

t-Based Confidence Interval for Paired Differences in Means

n/s

Ddt=

n

std

d

d/

0

2

10-26

Paired Differences Testing Rules

10-27

Example 10.6 and 10.7: The Repair Cost Comparison Case

2053.475033.

08.

0:

0:

3346.,2654.17

5033.447.28.

0

2

n/s

Ddt=

H

H

n

std

d

da

do

d/

10-28

10.4 Comparing Two Population Proportions by Using Large, Independent Samples

Select a random sample of size n1 from a population, and let p̂1 denote the proportion of units in this sample that fall into the category of interest

Select a random sample of size n2 from another population, and let p̂2 denote the proportion of units in this sample that fall into the same category of interest

Suppose that n1 and n2 are large enough n1·p1 ≥ 5, n1·(1 - p1) ≥ 5, n2·p2 ≥ 5, and n2·(1 –

p2) ≥ 5

10-29

Comparing Two Population Proportions Continued

Then the population of all possible values of p̂1 - p̂2

Has approximately a normal distribution if each of the sample sizes n1 and n2 is large

Has mean µp̂1 - p̂2 = p1 – p2

Has standard deviation 2

22

1

11 1121 n

pp

n

ppp̂p̂

10-30

Difference of Two Population Proportions

21 ˆˆ

021

2

22

1

11221

ˆˆ

ˆ1ˆˆ1ˆˆˆ

pp

Dppz=

n

pp

n

ppzpp

10-31

Example 10.9 and 10.10: The Advertising Media Case

2673.8

000,11

000,11

7145.17145.

0798.631.

11ˆ1ˆ

ˆˆˆˆ

1281.,2059.

000,1

202.798.

000,1

369.631.96.1798.631.

ˆ1ˆˆ1ˆˆˆ

21

021

ˆˆ

021

2

22

1

11221

21

z

nnpp

DppDppz=

n

pp

n

ppzpp

pp

10-32

10.5 Comparing Two Population Variances Using Independent Samples

Population 1 has variance σ12 and population 2

has variance σ22

The null hypothesis H0 is that the variances are the same H0: σ1

2 = σ22

The alternative is that one is smaller than the other That population has less variable measurements Suppose σ1

2 > σ22

More usual to normalize

Test H0: σ12/σ2

2 = 1 vs. σ12/σ2

2 > 1

10-33

Comparing Two Population Variances Using Independent Samples Continued

Reject H0 in favor of Ha if s12/s2

2 is significantly greater than 1

s12 is the variance of a random of size n1 from

a population with variance σ12

s22 is the variance of a random of size n2 from

a population with variance σ22

To decide how large s12/s2

2 must be to reject H0, describe the sampling distribution of s1

2/s22

The sampling distribution of s12/s2

2 is the F distribution

10-34

F Distribution

Figure 10.13

10-35

F Distribution

The F point F is the point on the horizontal axis under the curve of the F distribution that gives a right-hand tail area equal to The value of F depends on a (the size of the

right-hand tail area) and df1 and df2

Different F tables for different values of Tables A.5 for = 0.10 Tables A.6 for = 0.05 Tables A.7 for = 0.025 Tables A.8 for = 0.01

10-36


2544.1386

2.484

:

:

21

22

21

22

22

21

22

210

s

sF

H

H

a

Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Statistical Inferences Based on Two Samples Chapter 10.

Documents

slide slide

sample means

population variances

variances unknown slide

large slide

process slide

continued slide

variances equal slide