# SECTION 9.3 HYPOTHESIS TESTS FOR THE DIFFERENCE BETWEEN TWO MEANS: PAIRED SAMPLES Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for.

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SECTION 9.3 HYPOTHESIS TESTS FOR THE DIFFERENCE BETWEEN TWO MEANS: PAIRED SAMPLES Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Objectives 1. Perform a hypothesis test with matched pairs using the P- value method 2. Perform a hypothesis test with matched pairs using the critical value method 3. Construct confidence intervals with paired samples Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Perform a hypothesis test with matched pairs using the P-value method Objective 1 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Paired Samples A sample of eight automobiles were run to determine their mileage, in miles per gallon. Then each car was given a tune-up, and run again to measure the mileage a second time. The sample mean mileage was higher after tune-up. We would like to determine how strong the evidence is that the population mean mileage is higher after tune-up. These are paired samples, because each value before tune-up is paired with the value from the same car after tune-up. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Matched Pairs Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Notation Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Assumptions Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Hypothesis Test with Matched-Pair Data Using the P-Value Method Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Perform a hypothesis test with matched pairs using the critical value method Objective 2 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Testing a Hypothesis with Matched-Pair Data Using the Critical Value Method Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example For a sample of nine automobiles, the mileage (in 1000s of miles) at which the original front brake pads were worn to 10% of their original thickness was measured, as was the mileage at which the original rear brake pads were worn to 10% of their original thickness. The results are given below. The differences in the last line of the table are Rear − Front. Can you conclude that the mean time for the rear brake pads to wear out is longer than the mean time for the frontpads? Use the α = 0.05 significance level. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Construct confidence intervals with paired samples Objective 3 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Paired Samples Suppose we select sixteen volunteers and they are given a test in which they had to push a button in response to the appearance of an image on a screen. Their reaction times are measured. Then the subjects consumed enough alcohol to raise their blood alcohol level to 0.05%. They then took the reaction test again. Now, we have gathered two samples of data, a sample of reaction times before alcohol consumption, and a sample after alcohol consumption. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Notation We can compute the means of the two original samples as well as the mean of the sample of differences between each matched pair. The data for our experiment, along with the means, is presented in the table: 0.05%0%Difference 1102103 2100991 377698 4615011 58596-11 6502624 7957124 81151096 9645311 1098899 111071034 12442717 134750-3 1492100-8 1570664 1694868 Sample Mean81.374.86.5 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Confidence Interval Using Matched Pairs Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Confidence Interval for the Mean Difference Between Matched Pairs Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 0.05%0%Difference 1102103 2100991 377698 4615011 58596-11 6502624 7957124 81151096 9645311 1098899 111071034 12442717 134750-3 1492100-8 1570664 1694868 Sample Mean81.374.86.5 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Solution Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Matched Pairs and Margin of Error Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Matched Pairs and Margin of Error Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Do You Know… How to perform a hypothesis test with matched pairs using the P-value method? How to perform a hypothesis test with matched pairs using the critical value method? How to construct confidence intervals with paired samples? Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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