John Loucks St . Edward’s - PowerPoint Presentation

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John Loucks

St

. Edward’s

University

.

.

.

.

.......

SLIDES

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BY

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Chapter 10, Part A Inference About Means and Proportionswith Two Populations

Inferences About the Difference Between Two Population Means: s 1 and s 2 Known

Inferences About the Difference Between

Two Population Means: Matched Samples

Inferences About the Difference Between

Two Population Means:

s

1

and

s

2

Unknown

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Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Known

Interval Estimation of

m

1

–

m

2

Hypothesis Tests About

m

1

–

m

2

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Estimating the Difference BetweenTwo Population Means

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 .

Slide5

Expected Value

Sampling Distribution of

Standard Deviation (Standard Error)

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

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Interval Estimate

Interval Estimation of 1 - 2: s 1 and s 2 Known

where:

1 -



is the confidence coefficient

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Interval Estimation of 1 - 2: s 1 and s 2 Known

In a test of driving distance using a mechanicaldriving device, a sample of Par golf balls wascompared with a sample of golf balls made by Rap,Ltd., a competitor. The sample statistics appear onthe next slide.

Par, Inc. is a

manufacturer of

golf equipment andhas developed a new golf ball that has been designed to provide “extra distance.”

Example: Par, Inc.

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Example: Par, Inc.

Interval Estimation of

1 - 2: s 1 and s 2 Known

Sample Size

Sample Mean

Sample #1Par, Inc.

Sample #2Rap, Ltd.

120 balls 80 balls

275 yards 258 yards

Based on data from previous driving distance

tests, the two population standard deviations are

known with

s

1

= 15 yards and

s

2

= 20 yards.

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Interval Estimation of 1 - 2: s 1 and s 2 Known

Example: Par, Inc.

Let us develop a 95% confidence interval estimateof the difference between the mean driving distances ofthe two brands of golf ball.

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Estimating the Difference BetweenTwo Population Means

m

1 – m2 = difference between the mean distances

x

1

- x2 = Point Estimate of m1 – m2

Population 1

Par, Inc. Golf Balls

m

1

= mean driving

distance of Par

golf balls

Population 2

Rap, Ltd. Golf Balls

m

2

= mean driving

distance of Rapgolf balls

Simple random sample

of

n

2 Rap golf ballsx2 = sample mean distance for the Rap golf balls

Simple random sample

of

n

1

Par golf ballsx1 = sample mean distance for the Par golf balls

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Point Estimate of 1 - 2

Point estimate of 1 - 2 =

where:



1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls

= 275

-

258

= 17 yards

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Interval Estimation of



1 - 2: 1 and  2 Known

We are 95% confident that the difference between

the mean driving distances of Par, Inc. balls and Rap,

Ltd. balls is 11.86 to 22.14 yards.

17

+

5.14 or 11.86 yards to 22.14 yards

Slide13

Hypothesis Tests About

m

1

- m 2:s 1 and s 2 Known

Hypotheses

Left-tailed

Right-tailed

Two-tailed

Test Statistic

Slide14

Example: Par, Inc.

Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known

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

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H0: 1 - 2 < 0 Ha: 1 - 2 > 0

where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls

1. Develop the hypotheses.

p

–Value and Critical Value Approaches

Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known

2. Specify the level of significance.

a

= .01

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3. Compute the value of the test statistic.

Hypothesis Tests About

m

1 - m 2:s 1 and s 2 Known

p –Value and Critical Value Approaches

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p –Value Approach

4. Compute the

p–value.

For

z = 6.49, the p –value < .0001.

Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known

5. Determine whether to reject

H0.

Because

p–value < a = .01, we reject H0.

At the .01 level of significance, the sample evidence

indicates the mean driving distance of Par, Inc. golf

balls is greater than the mean driving distance of Rap,

Ltd. golf balls.

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Hypothesis Tests About m 1 - m 2:s 1 and s 2 Known

5. Determine whether to reject

H0.

Because z = 6.49 > 2.33, we reject H0.

Critical Value Approach

For

a

= .01, z.01 = 2.33

4. Determine the critical value and rejection rule.

Reject

H0 if z > 2.33

The sample evidence indicates the mean driving

distance of Par, Inc. golf balls is greater than the mean

driving distance of Rap, Ltd. golf balls.

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Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Unknown

Interval Estimation of

m

1

–

m

2

Hypothesis Tests About

m

1

–

m

2

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Interval Estimation of 1 - 2:s 1 and s 2 Unknown

When s 1 and s 2 are unknown, we will:

replace za/2 with ta/2.

use the sample standard deviations s1 and s2 as estimates of s 1 and s 2 , and

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Where the degrees of freedom for

t

a

/2

are:

Interval Estimation of

1 - 2:s 1 and s 2 Unknown

Interval Estimate

Slide22

Example: Specific Motors

Difference Between Two Population Means:s 1 and s 2 Unknown

Specific Motors of

Detroit has

developed a

new

Automobile known

as the M car. 24 M

cars and

28

J

cars

(from Japan) were

road tested

to compare

miles-

per-gallon

(mpg) performance.

The

sample

statistics

are

shown on the next slide.

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Difference Between Two Population Means:

s 1 and s 2 Unknown

Example: Specific Motors

Sample Size

Sample Mean

Sample Std. Dev.

Sample #1M Cars

Sample #2J Cars

24 cars 28 cars

29.8 mpg 27.3 mpg

2.56 mpg 1.81 mpg

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Difference Between Two Population Means:s 1 and s 2 Unknown

Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile.

Example: Specific Motors

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Point estimate of 1 - 2 =

Point Estimate of

m

1 - m 2

where: 1 = mean miles-per-gallon for the population of M cars 2 = mean miles-per-gallon for the population of J cars

= 29.8 - 27.3

= 2.5 mpg

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Interval Estimation of m 1 - m 2:s 1 and s 2 Unknown

The degrees of freedom for ta/2 are:

With

a

/2 = .05 and

df = 24, ta/2 = 1.711

Slide27

Interval Estimation of m 1 - m 2:s 1 and s 2 Unknown

We are 90% confident that the difference between

the miles-per-gallon performances of M cars and J cars

is 1.431 to 3.569 mpg.

2.5

+

1.069 or 1.431 to 3.569 mpg

Slide28

Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown

Hypotheses

Left-tailed

Right-tailed

Two-tailed

Test Statistic

Slide29

Example: Specific Motors

Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown

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?

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H0: 1 - 2 < 0 Ha: 1 - 2 > 0

where: 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars

1. Develop the hypotheses.

p

–Value and Critical Value Approaches

Hypothesis Tests About

m

1

-

m

2

:

s

1

and

s

2

Unknown

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2. Specify the level of significance.

3. Compute the value of the test statistic.

a

= .05

p

–Value and Critical Value Approaches

Hypothesis Tests About

m 1 - m 2:s 1 and s 2 Unknown

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Hypothesis Tests About m 1 - m 2:s 1 and s 2 Unknown

p –Value Approach

4. Compute the

p

–value.

The degrees of freedom for ta are:

Because t = 4.003 > t.005 = 1.683, the p–value < .005.

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5. Determine whether to reject

H0.

We are at least 95% confident that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars?.

p –Value Approach

Because

p–value < a = .05, we reject H0.

Hypothesis Tests About

m

1

-

m

2

:

s

1

and

s

2

Unknown

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4. Determine the critical value and rejection rule.

Critical Value Approach

Hypothesis Tests About

m

1

- m 2:s 1 and s 2 Unknown

For a = .05 and df = 41, t.05 = 1.683

Reject H0 if t > 1.683

5. Determine whether to reject

H0.

Because 4.003 > 1.683, we reject H0.

We are at least 95% confident that the miles-per-gallon (

mpg

) performance of M cars is greater than the miles-per-gallon performance of J cars?.

Slide35

With a

matched-sample design each sampled item provides a pair of data values.

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.

Inferences About the Difference Between

Two Population Means: Matched Samples

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Example: Express Deliveries

Inferences About the Difference BetweenTwo Population Means: Matched Samples

A Chicago-based firm has documents that mustbe quickly distributed to district offices throughout the U.S. The firm must decide between two deliveryservices, UPX (United Parcel Express) and INTEX(International Express), to transport its documents.

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Example: Express Deliveries

Inferences About the Difference BetweenTwo Population Means: Matched Samples

In testing the delivery

times of

the two services

,

the

firm

sent two

reports to a random

sample of

its

district

offices with

one report

carried by UPX and

the other

report carried by INTEX. Do the data

on

the next

slide indicate a difference in mean delivery

times for the two services? Use a .05 level of

significance.

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32

30191615181410 716

25241515131515 8 911

UPX

INTEX

Difference

District Office

SeattleLos AngelesBostonClevelandNew YorkHoustonAtlantaSt. LouisMilwaukeeDenver

Delivery Time (Hours)

7

6

4

1

2 3 -1 2 -2 5

Inferences About the Difference Between

Two Population Means: Matched Samples

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H0: d = 0 Ha: d 

Let d = the mean of the difference values for the two delivery services for the population of district offices

1. Develop the hypotheses.

Inferences About the Difference Between

Two Population Means: Matched Samples

p

–Value and Critical Value Approaches

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2. Specify the level of significance.

a

= .05

Inferences About the Difference Between

Two Population Means: Matched Samples

p –Value and Critical Value Approaches

3. Compute the value of the test statistic.

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5. Determine whether to reject

H0.

We are at least 95% confident that there is a difference in mean delivery times for the two services?

4. Compute the

p

–value.

For

t = 2.94 and df = 9, the p–value is between.02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.)

Because p–value < a = .05, we reject H0.

Inferences About the Difference BetweenTwo Population Means: Matched Samples

p

–Value Approach

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4. Determine the critical value and rejection rule.

Inferences About the Difference Between

Two Population Means: Matched Samples

Critical Value Approach

For

a = .05 and df = 9, t.025 = 2.262.

Reject H0 if t > 2.262

5. Determine whether to reject

H0.

Because t = 2.94 > 2.262, we reject H0.

We are at least 95% confident that there is a difference in mean delivery times for the two services?

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End of Chapter 10Part A