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Slides Prepared by JOHN S. LOUCKS St. Edward’s University. Chapter 10 Statistical Inferences about Means and Proportions for Two Populations. Estimation of the Difference Between the Means of Two Populations: Independent Samples Hypothesis Tests about the Difference Between the

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JOHN S. LOUCKS

St. Edward’s University

Chapter 10 Statistical Inferences about Means and Proportions for Two Populations

• 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

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

Point Estimator of the Difference between Populations: Independent Samplesthe 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 .

Sampling Distribution of Populations: Independent Samples

• 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

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

• Interval Estimate with 1 and 2 Assumed Known

where:

1 -  is the confidence coefficient

• Interval Estimate with 1 and 2 Estimated by s1 and s2

where:

Example: Par, Inc. Populations: Independent Samples

• 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.

Example: Par, Inc. Populations: Independent Samples

• 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 balls

Mean = 235 yards = 218 yards

Standard Dev. s1 = 15 yards s2 = 20 yards

Example: Par, Inc. Populations: Independent Samples

• Point Estimate of the Difference Between Two Population Means

1 = mean distance for the population of

Par, Inc. golf balls

2 = mean distance for the population of

Rap, Ltd. golf balls

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

Simple random sample Populations: Independent Samples

of n1 Par golf balls

x1 = sample mean distance

for sample of Par golf ball

Simple random sample

of n2 Rap golf balls

x2 = sample mean distance

for sample of Rap golf ball

x1 - x2 = Point Estimate of m1 –m2

Point Estimator of the Difference

between the Means of Two Populations

Population 1

Par, Inc. Golf Balls

m1 = mean driving

distance of Par

golf balls

Population 2

Rap, Ltd. Golf Balls

m2 = mean driving

distance of Rap

golf balls

m1 –m2= difference between

the mean distances

Example: Par, Inc. Populations: Independent Samples

• 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Estimated by s1 and s2

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.

Using Excel to Develop an Interval Estimate Populations: Independent Samples

of m1 – m2: Large-Sample Case

• Formula Worksheet

Note: Rows 16-121 are not shown.

Using Excel to Develop an Interval Estimate Populations: Independent Samples

of m1 – m2: Large-Sample Case

• Value Worksheet

Note: Rows 16-121 are not shown.

Interval Estimate of Populations: Independent Samples1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30)

• Interval Estimate with  2 Assumed Known

where:

Interval Estimate of Populations: Independent Samples1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30)

• Interval Estimate with 1 and 2 Estimated by s1 and s2

where:

Example: Specific Motors Populations: Independent Samples

Specific Motors of Detroit has developed a new

automobile 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 CarsJ 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

Example: Specific Motors Populations: Independent Samples

• Point Estimate of the Difference Between Two Population Means

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

M cars

2 = mean miles-per-gallon for the population of

J cars

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

Example: Specific Motors Populations: Independent Samples

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

We 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 degrees

of 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 Populations: Independent Samples

• 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 the

mean mpg ratings of the two car types is from 0.3 to 4.7 mpg (with the M car having the higher mpg).

Using Excel to Develop an Interval Estimate Populations: Independent Samples

of m1 – m2: Small-Sample Case

• Formula Worksheet

Using Excel to Develop an Interval Estimate Populations: Independent Samples

of m1 – m2: Small-Sample Case

• Value Worksheet

Hypothesis Tests about the Difference Populations: Independent Samplesbetween 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

Example: Par, Inc. Populations: Independent Samples

• 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. Populations: Independent Samples

• 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 balls

Mean = 235 yards = 218 yards

Standard Dev. s1 = 15 yards s2 = 20 yards

Example: Par, Inc. Populations: Independent Samples

• 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 balls

2 = mean distance for the population of Rap, Ltd.

golf balls

• HypothesesH0: 1 - 2< 0

Ha: 1 - 2 > 0

Example: Par, Inc. Populations: Independent Samples

• Hypothesis Tests about the Difference between the Means of Two Populations: Large-Sample Case

• Rejection RuleReject 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.

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Large Sample Case

• Excel’s “z-Test: Two Sample for Means” Tool

Step 1Select the Tools pull-down menu

Step 2Choose the Data Analysis option

Step 3 Choose z-Test: Two Sample for Means

from the list of Analysis Tools

… continued

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Large Sample Case

• Excel’s “z-Test: Two Sample for Means” Tool

Step 4When the z-Test: Two Sample for Means

dialog box appears:

Enter A1:A121 in the Variable 1 Range box

Enter B1:B81 in the Variable 2 Range box

Enter 0 in the Hypothesized Mean Difference box

Enter 225 in the Variable 1 Variance (known) box

Enter 400 in the Variable 2 Variance (known) box

… continued

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Large Sample Case

• Excel’s “z-Test: Two Sample for Means” Tool

Step 4 (continued)

Select Labels

Enter .01 in the Alpha box

Select Output Range

Enter D4 in the Output Range box

(Any upper left-hand corner cell indicating

where the output is to begin may be entered)

Click OK

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Large Sample Case

• Value Worksheet

Note: Rows 16-121 are not shown.

Example: Specific Motors Populations: Independent Samples

• 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 Populations: Independent Samples

• 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:

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Small Sample Case

• Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool

Step 1Select the Tools pull-down menu

Step 2Choose the Data Analysis option

Step 3 Choose t-Test: Two Sample Assuming Equal Variances from the list of Analysis Tools

… continued

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Small Sample Case

• Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool

Step 4When the t-Test: Two Sample Assuming Equal Variances dialog box appears:

Enter A1:A13 in the Variable 1 Range box

Enter B1:B9 in the Variable 2 Range box

Enter 0 in the Hypothesized Mean Difference box

… continued

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Small Sample Case

• Excel’s “t-Test: Two Sample Assuming Equal Variances” Tool

Step 4 (continued)

Select Labels

Enter .01 in the Alpha box

Select Output Range

Enter D1 in the Output Range box

(Any upper left-hand corner cell indicating

where the output is to begin may be entered)

Click OK

Using Excel to Conduct a Hypothesis Test Populations: Independent Samplesabout m1 – m2: Small Sample Case

• Value Worksheet

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.

Example: Express Deliveries Populations: Matched Samples

• 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?

Example: Express Deliveries Populations: Matched Samples

Delivery Time (Hours)

District OfficeUPXINTEXDifference

Seattle 32 25 7

Los Angeles 30 24 6

Boston 19 15 4

Cleveland 16 15 1

New York 15 13 2

Houston 18 15 3

Atlanta 14 15 -1

St. Louis 10 8 2

Milwaukee 7 9 -2

Denver 16 11 5

Example: Express Deliveries Populations: Matched Samples

• 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

• HypothesesH0: 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 Populations: Matched Samples

• Inference about the Difference between the Means of Two Populations: Matched Samples

• ConclusionReject H0.

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

Using Excel to Conduct a Hypothesis Test Populations: Matched Samplesabout m1 – m2: Matched Samples

• Excel’s “t-Test: Paired Two Sample for Means” Tool

Step 1Select the Tools pull-down menu

Step 2Choose the Data Analysis option

Step 3 Choose t-Test: Paired Two Sample for Means

from the list of Analysis Tools

… continued

Using Excel to Conduct a Hypothesis Test Populations: Matched Samplesabout m1 – m2: Matched Samples

• Excel’s “t-Test: Paired Two Sample for Means” Tool

Step 4When the t-Test: Paired Two Sample for Means

dialog box appears:

Enter B1:B11 in the Variable 1 Range box

Enter C1:C11 in the Variable 2 Range box

Enter 0 in the Hypothesized Mean Difference box

Select Labels

Enter .05 in the Alpha box

Select Output Range

Enter E2 (your choice) in the Output Range box

Click OK

Using Excel to Conduct a Hypothesis Test Populations: Matched Samplesabout m1 – m2: Matched Samples

• Value Worksheet

Inferences about the Difference between the Proportions of Two Populations

• Sampling Distribution of

• Interval Estimation of p1 - p2

• Hypothesis Tests about p1 - p2

Sampling Distribution of Two Populations

• Expected Value

• Standard Deviation

• Distribution Form

If the sample sizes are large (n1p1, n1(1 - p1), n2p2,

and n2(1 - p2) are all greater than or equal to 5), the

sampling distribution of can be approximated

by a normal probability distribution.

Interval Estimation of Two Populationsp1 - p2

• Interval Estimate

• Point Estimator of

Example: MRA Two Populations

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?

Example: MRA Two Populations

• 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

Example: MRA Two Populations

• Interval Estimate of p1 - p2: Large-Sample Case

For = .05, z.025 = 1.96:

.08 + 1.96(.0510)

.08 + .10

or -.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.

Using Excel to Develop Two Populationsan Interval Estimate of p1 – p2

• Formula Worksheet

Note: Rows 16-251 are not shown.

Using Excel to Develop Two Populationsan Interval Estimate of p1 – p2

• Value Worksheet

Note: Rows 16-251 are not shown.

Hypothesis Tests about Two Populationsp1 - p2

• Hypotheses

H0: p1 - p2< 0

Ha: p1 - p2 > 0

• Test statistic

• Point Estimator of where p1 = p2

where:

Example: MRA Two Populations

• 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

• HypothesesH0: p1 - p2< 0

Ha: p1 - p2 > 0

Example: MRA Two Populations

• Hypothesis Tests about p1 - p2

• Rejection RuleReject H0 if z > 1.645

• Test Statistic

• ConclusionDo not reject H0.

Using Excel to Conduct Two Populationsa Hypothesis Test about p1 – p2

• Formula Worksheet

Note: Rows 17-251 are not shown.

Using Excel to Conduct Two Populationsa Hypothesis Test about p1 – p2

• Value Worksheet

Note: Rows 17-251 are not shown.

End of Chapter 10 Two Populations