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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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Slides Prepared by

JOHN S. LOUCKS

St. Edward’s University


Chapter 10 statistical inferences about means and proportions for two populations
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
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 the means of two populations
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
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 1 2 large sample case n 1 30 and n 2 30
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
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 inc1
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 inc2
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 inc3
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 1 2 small sample case n 1 30 and or n 2 30
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 1 2 small sample case n 1 30 and or n 2 301
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
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 motors1
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 motors2
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 motors3
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 between the means of two populations independent samples
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 inc4
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 inc5
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 inc6
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 inc7
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 about m 1 m 2 large sample case
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 about m 1 m 2 large sample case1
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 about m 1 m 2 large sample case2
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 about m 1 m 2 large sample case3
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 motors4
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 motors5
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 about m 1 m 2 small sample case
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 about m 1 m 2 small sample case1
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 about m 1 m 2 small sample case2
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 about m 1 m 2 small sample case3
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
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
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 deliveries1
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 deliveries2
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 deliveries3
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 about m 1 m 2 matched samples
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 about m 1 m 2 matched samples1
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 about m 1 m 2 matched samples2
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
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 of1
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 p 1 p 2
Interval Estimation of Two Populationsp1 - p2

  • Interval Estimate

  • Point Estimator of


Example mra
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 mra1
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 mra2
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 an interval estimate of p 1 p 2
Using Excel to Develop Two Populationsan Interval Estimate of p1 – p2

  • Formula Worksheet

Note: Rows 16-251 are not shown.


Using excel to develop an interval estimate of p 1 p 21
Using Excel to Develop Two Populationsan Interval Estimate of p1 – p2

  • Value Worksheet

Note: Rows 16-251 are not shown.


Hypothesis tests about p 1 p 2
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 mra3
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 mra4
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 a hypothesis test about p 1 p 2
Using Excel to Conduct Two Populationsa Hypothesis Test about p1 – p2

  • Formula Worksheet

Note: Rows 17-251 are not shown.


Using excel to conduct a hypothesis test about p 1 p 21
Using Excel to Conduct Two Populationsa Hypothesis Test about p1 – p2

  • Value Worksheet

Note: Rows 17-251 are not shown.


End of chapter 10
End of Chapter 10 Two Populations


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