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Slides Prepared byJOHN S. LOUCKS

St. Edward’s University

© 2002 South-Western /Thomson Learning

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Chapter 10 Comparisons Involving Means

1 = 2 ?

ANOVA

Estimation of the Difference Between the Means of

Two Populations: Independent Samples Hypothesis Tests about the Difference

between the Means of Two Populations: Independent

Samples Inferences about the Difference between the

Means of Two Populations: Matched Samples Inferences about the Difference between the Proportions of Two Populations:

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Estimation of the Difference Between the Means

of Two Populations: Independent Samples Point Estimator of the Difference between the

Means of Two Populations Sampling Distribution Interval Estimate of Large-Sample Case Interval Estimate of Small-Sample Case

x x1 2

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Point Estimator of the Difference Betweenthe Means of Two Populations

Let 1 equal the mean of population 1 and 2 equal the mean of population 2.

The difference between the two population means is 1 - 2.

To estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2.

Let equal the mean of sample 1 and equal the mean of sample 2.

The point estimator of the difference between the means of the populations 1 and 2 is .

21 xx

x1 x2

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Properties of the Sampling Distribution of • Expected Value

• Standard Deviation

where: 1 = standard deviation of population 1

2 = standard deviation of population 2

n1 = sample size from population 1

n2 = sample size from population 2

Sampling Distribution of x x1 2

x x1 2

E x x( )1 2 1 2

x x n n1 2

12

1

22

2

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Interval Estimate with 1 and 2 Known

where:1 - is the confidence coefficient

Interval Estimate with 1 and 2 Unknown

where:

Interval Estimate of 1 - 2:Large-Sample Case (n1 > 30 and n2 > 30)

x x z x x1 2 2 1 2 /

x x z sx x1 2 2 1 2 /

ssn

snx x1 2

12

1

22

2

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Example: Par, Inc.

Interval Estimate of 1 - 2: Large-Sample Case

Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor.

The sample statistics appear on the next slide.

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Example: Par, Inc.

Interval Estimate of 1 - 2: Large-Sample Case

• Sample Statistics

Sample #1 Sample #2

Par, Inc. Rap, Ltd.Sample Size n1 = 120 balls n2 = 80 ballsMean = 235 yards = 218 yardsStandard Dev. s1 = 15 yards s2 = 20 yards

x1 2x

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Point Estimate of the Difference Between Two Population Means

1 = mean distance for the population of

Par, Inc. golf balls2 = mean distance for the population of

Rap, Ltd. golf balls

Point estimate of 1 - 2 = = 235 - 218 = 17 yards.

x x1 2

Example: Par, Inc.

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Point Estimator of the Difference Between the Means of Two Populations

Population 1Par, Inc. Golf Balls

m1 = mean driving distance of Par

golf balls

Population 1Par, Inc. Golf Balls

m1 = mean driving distance of Par

golf balls

Population 2Rap, Ltd. Golf Balls

m2 = mean driving distance of Rap

golf balls

Population 2Rap, Ltd. Golf Balls

m2 = mean driving distance of Rap

golf balls

m1 – m2 = difference between the mean distances

Simple random sample of n1 Par golf balls

x1 = sample mean distancefor sample of Par golf ball

Simple random sample of n1 Par golf balls

x1 = sample mean distancefor sample of Par golf ball

Simple random sample of n2 Rap golf balls

x2 = sample mean distancefor sample of Rap golf ball

Simple random sample of n2 Rap golf balls

x2 = sample mean distancefor sample of Rap golf ball

x1 - x2 = Point Estimate of m1 – m2

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95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown

Substituting the sample standard deviations for the population standard deviation:

= 17 + 5.14 or 11.86 yards to 22.14 yards.We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of 11.86 to 22.14 yards.

x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

Example: Par, Inc.

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Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 <

30) Interval Estimate with 2 Known

where:

x x z x x1 2 2 1 2 /

x x n n1 2

2

1 2

1 1 ( )

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Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 <

30) Interval Estimate with 2 Unknown

where:

x x t sx x1 2 2 1 2 /

sn s n s

n n2 1 1

22 2

2

1 2

1 12

( ) ( )s sn nx x1 2

2

1 2

1 1 ( )

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Example: Specific Motors

Specific Motors of Detroit has developed a newautomobile known as the M car. 12 M cars and 8

J cars(from Japan) were road tested to compare miles-

per-gallon (mpg) performance. The sample statistics

are:

Sample #1 Sample #2

M Cars J Cars

Sample Size n1 = 12 cars n2 = 8 cars

Mean = 29.8 mpg = 27.3 mpg

Standard Deviation s1 = 2.56 mpg s2 = 1.81 mpg

x2x1

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Point Estimate of the Difference Between Two Population Means

1 = mean miles-per-gallon for the population of

M cars2 = mean miles-per-gallon for the population of

J cars

Point estimate of 1 - 2 = = 29.8 - 27.3 = 2.5 mpg.

x x1 2

Example: Specific Motors

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95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample CaseWe will make the following assumptions:• The miles per gallon rating must be

normally distributed for both the M car and the J car.• The variance in the miles per gallon rating

must be the same for both the M car and the J

car.Using the t distribution with n1 + n2 - 2 = 18

degreesof freedom, the appropriate t value is t.025 =

2.101.We will use a weighted average of the two sample

variances as the pooled estimator of 2.

Example: Specific Motors

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95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case

= 2.5 + 2.2 or .3 to 4.7 miles per gallon.We are 95% confident that the difference between themean mpg ratings of the two car types is from .3 to 4.7 mpg (with the M car having the higher mpg).

sn s n s

n n2 1 1

22 2

2

1 2

2 21 12

11 2 56 7 1 8112 8 2

5 28

( ) ( ) ( . ) ( . ).

x x t sn n1 2 025

2

1 2

1 12 5 2 101 5 28

112

18

. ( ) . . . ( )

Example: Specific Motors

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Hypothesis Tests About the Difference Between the Means of Two Populations:

Independent Samples Hypotheses

H0: 1 - 2 < 0 H0: 1 - 2 > 0 H0: 1

- 2 = 0

Ha: 1 - 2 > 0 Ha: 1 - 2 < 0 Ha: 1

- 2 0

Test Statistic Large-Sample

Small-Sample

zx x

n n

( ) ( )1 2 1 2

12

1 22

2

tx x

s n n

( ) ( )

( )1 2 1 2

21 21 1

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Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide.

Example: Par, Inc.

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Example: Par, Inc.

Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case• Sample Statistics

Sample #1 Sample #2

Par, Inc. Rap, Ltd.Sample Size n1 = 120 balls n2 = 80 ballsMean = 235 yards = 218 yardsStandard Dev. s1 = 15 yards s2 = 20 yards

x1 x2

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Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case

Can we conclude, using a .01 level of significance, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls?1 = mean distance for the population of Par, Inc.

golf balls2 = mean distance for the population of Rap, Ltd.

golf balls• HypothesesH0: 1 - 2 < 0

Ha: 1 - 2 > 0

Example: Par, Inc.

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Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case• Rejection Rule Reject H0 if z > 2.33

• Conclusion Reject H0. We are at least 99%

confident that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls.

zx x

n n

( ) ( ) ( )

( ) ( ) ..1 2 1 2

12

1

22

2

2 2

235 218 0

15120

2080

172 62

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Example: Par, Inc.

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Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case

Can we conclude, using a .05 level of significance, that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars?1 = mean mpg for the population of M cars

2 = mean mpg for the population of J cars

• HypothesesH0: 1 - 2 < 0

Ha: 1 - 2 > 0

Example: Specific Motors

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Example: Specific Motors

Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case• Rejection Rule

Reject H0 if t > 1.734

(a = .05, d.f. = 18)

• Test Statistic

where:

tx x

s n n

( ) ( )

( )1 2 1 2

21 21 1

2 22 1 1 2 2

1 2

( 1) ( 1)

2

n s n ss

n n

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Inference About the Difference Between the Means of Two Populations: Matched

Samples With a matched-sample design each sampled

item provides a pair of data values. The matched-sample design can be referred to

as blocking. This design often leads to a smaller sampling

error than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.

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Example: Express Deliveries

Inference About the Difference Between the Means of Two Populations: Matched Samples

A Chicago-based firm has documents that must be quickly distributed to district offices throughout the U.S. The firm must decide between two delivery services, UPX (United Parcel Express) and INTEX (International Express), to transport its documents. In testing the delivery times of the two services, the firm sent two reports to a random sample of ten district offices with one report carried by UPX and the other report carried by INTEX.

Do the data that follow indicate a difference in mean delivery times for the two services?

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Delivery Time (Hours)District Office UPX INTEX DifferenceSeattle 32 25 7Los Angeles 30 24 6Boston 19 15 4Cleveland 16 15 1New York 15 13 2Houston 18 15 3Atlanta 14 15 -1St. Louis 10 8 2Milwaukee 7 9 -2Denver 16 11 5

Example: Express Deliveries

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Inference About the Difference Between the Means of Two Populations: Matched Samples Let d = the mean of the difference values for the two delivery services for the population of district offices

• Hypotheses H0: d = 0, Ha: d • Rejection Rule

Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With = .05, t.025 = 2.262 (9 degrees of freedom).

Reject H0 if t < -2.262 or if t > 2.262

Example: Express Deliveries

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Inference About the Difference Between the Means of Two Populations: Matched Samples

• Conclusion Reject H0.

There is a significant difference between the mean delivery times for the two services.

ddni ( ... )

.7 6 5

102 7

sd dndi

( ) ..

2

176 1

92 9

tds n

d

d

2 7 02 9 10

2 94.

..

Example: Express Deliveries

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Inferences About the Difference Between the Proportions of Two

Populations Sampling Distribution of Interval Estimation of p1 - p2

Hypothesis Tests about p1 - p2

p p1 2

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Expected Value

Standard Deviation

Distribution FormIf the sample sizes are large (n1p1, n1(1 -

p1), n2p2,and n2(1 - p2) are all greater than or equal to 5), thesampling distribution of can be approximatedby a normal probability distribution.

Sampling Distribution of p p1 2

E p p p p( )1 2 1 2

p pp pn

p pn1 2

1 1

1

2 2

2

1 1 ( ) ( )

p p1 2

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Interval Estimation of p1 - p2

Interval Estimate

Point Estimator of

p p z p p1 2 2 1 2 /

p p1 2

sp pn

p pnp p1 2

1 1

1

2 2

2

1 1 ( ) ( )

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Example: MRA

MRA (Market Research Associates) is conducting research to evaluate the effectiveness of a client’s new advertising campaign. Before the new campaign began, a telephone survey of 150 households in the test market area showed 60 households “aware” of the client’s product. The new campaign has been initiated with TV and newspaper advertisements running for three weeks. A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product.

Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

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Example: MRA

Point Estimator of the Difference Between the Proportions of Two Populations

p1 = proportion of the population of households

“aware” of the product after the new campaign

p2 = proportion of the population of households

“aware” of the product before the new campaign = sample proportion of households “aware” of the

product after the new campaign = sample proportion of households “aware” of the

product before the new campaign

p p p p1 2 1 2120250

60150

48 40 08 . . .

p1

p2

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Example: MRA

Interval Estimate of p1 - p2: Large-Sample Case

For = .05, z.025 = 1.96:

.08 + 1.96(.0510) .08 + .10or -.02 to +.18

• Conclusion At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is -.02 to +.18.

. . .. (. ) . (. )

48 40 1 9648 52

25040 60

150

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Hypothesis Tests about p1 - p2

Hypotheses H0: p1 - p2 < 0

Ha: p1 - p2 > 0 Test statistic

Point Estimator of where p1 = p2

where:

zp p p p

p p

( ) ( )1 2 1 2

1 2

s p p n np p1 21 1 11 2 ( )( )

pn p n pn n

1 1 2 2

1 2

p p1 2

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Example: MRA

Hypothesis Tests about p1 - p2

Can we conclude, using a .05 level of significance, that the proportion of households aware of the client’s product increased after the new advertising campaign?p1 = proportion of the population of households

“aware” of the product after the new campaign

p2 = proportion of the population of households

“aware” of the product before the new campaign • Hypotheses H0: p1 - p2 < 0

Ha: p1 - p2 > 0

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Example: MRA

Hypothesis Tests about p1 - p2

• Rejection Rule Reject H0 if z > 1.645

• Test Statistic

• Conclusion Do not reject H0.

p

250 48 150 40250 150

180400

45(. ) (. )

.

sp p1 245 55 1

2501150 0514 . (. )( ) .

z (. . ).

..

.48 40 0

051408

05141 56

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End of Chapter 10