Paired Samples. Lecture 37 Sections 11.1, 11.2, Robb T. Koether. Hampden-Sydney College. Mon, Apr 2, 2012
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1 Paired Samples Lecture 37 Sections 11.1, 11.2, 11.3 Robb T. Koether Hampden-Sydney College Mon, Apr 2, 2012 Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
2 Outline 1 Dependent Samples (Paired Data) 2 Assignment Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
3 Outline 1 Dependent Samples (Paired Data) 2 Assignment Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
4 Dependent Samples (Paired Data) Let the pairs be denoted (x 1, x 2 ). Let d = x 2 x 1. We will study the case where d has a normal distribution. Let µ D denote the mean of this distribution and σ D denote the standard deviation. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
5 The only null hypothesis for µ D that we will consider is H 0 : µ D = 0. We will consider any of the three alternatives H 1 : µ D < 0. H 1 : µ D > 0. H 1 : µ D 0. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
6 For large samples, the test statistic is (or z) t = d 0 s D / n. For small samples it is necessary that d have a normal distribution. Then the test statistic is t = d 0 s D / n. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
7 Example ( ) Suppose that a group of 10 students take a math placement test. Let the variable x 1 represent their scores on that test. Then they are given an Algebra refresher course and they retake the placement test. Let the variable x 2 represent their scores on the retest. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
8 Example ( ) The following table shows the results Student 1st Score (x 1 ) 2nd Score (x 2 ) Difference (d) Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
9 Example ( ) The following table shows the results Student 1st Score (x 1 ) 2nd Score (x 2 ) Difference (d) Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
10 Example ( ) Test the hypothesis, at the 10% level, that the refresher course improved their grades on the placement test. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
11 Example ( ) (1) Let x 1 be the first test score, let x 2 be the second test score, and let d = x 2 x 1. Then the hypotheses are H 0 : µ D = 0. H 1 : µ D > 0. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
12 Example ( ) (1) Let x 1 be the first test score, let x 2 be the second test score, and let d = x 2 x 1. Then the hypotheses are H 0 : µ D = 0. H 1 : µ D > 0. (2) α = Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
13 Example ( ) (1) Let x 1 be the first test score, let x 2 be the second test score, and let d = x 2 x 1. Then the hypotheses are H 0 : µ D = 0. H 1 : µ D > 0. (2) α = (3) Let t = d 0 s D / n. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
14 Example ( ) (4) Compute the value of the test statistic. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
15 Example ( ) (4) Compute the value of the test statistic. Enter the x 1 values into L 1 and the x 2 values into L 2. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
16 Example ( ) (4) Compute the value of the test statistic. Enter the x 1 values into L 1 and the x 2 values into L 2. Evaluate the difference L 2 L 1 and store it in L 3. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
17 Example ( ) (4) Compute the value of the test statistic. Enter the x 1 values into L 1 and the x 2 values into L 2. Evaluate the difference L 2 L 1 and store it in L 3. Use 1-Var Stats L 3 to get d and s D. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
18 Example ( ) (4) Compute the value of the test statistic. Enter the x 1 values into L 1 and the x 2 values into L 2. Evaluate the difference L 2 L 1 and store it in L 3. Use 1-Var Stats L 3 to get d and s D. We find that d = 3 and s D = Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
19 Example ( ) (4) Compute the value of the test statistic. Enter the x 1 values into L 1 and the x 2 values into L 2. Evaluate the difference L 2 L 1 and store it in L 3. Use 1-Var Stats L 3 to get d and s D. We find that d = 3 and s D = Then 3 t = 5.354/ 10 = = Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
20 Example ( ) (5) p-value = tcdf(1.772,e99,9) = Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
21 Example ( ) (5) p-value = tcdf(1.772,e99,9) = (6) Reject H 0. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
22 Example ( ) (5) p-value = tcdf(1.772,e99,9) = (6) Reject H 0. (7) Students scores on the placement test are higher after taking the Algebra refresher course. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
23 Quiz Average vs. Test Average Example (Quiz Average vs. Test Average) Is a student s quiz average higher than his test average, on the average? Here are the data from a past semester. Test Avg. Quiz Avg. Diff Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
24 Quiz Average vs. Test Average Example (Quiz Average vs. Test Average) Is a student s quiz average higher than his test average, on the average? Here are the data from a past semester. Test Avg. Quiz Avg. Diff Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
25 Outline 1 Dependent Samples (Paired Data) 2 Assignment Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
26 Assignment Homework Read Sections 11.1, 11.2, 11.3, pages Let s Do It! 11.1, 11.2, Exercises 1-8, page 676. Exercises 9-14, page 689. Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, / 17
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