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Tuesday, October 22 Interval estimation. Independent samples t-test for the difference between two means. Matched samples t-test

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Tuesday, October 23 Interval estimation. Independent samples t-test for the difference between two means. Matched samples t-test

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Interval Estimation (a.k.a. confidence interval) Is there a range of possible values for that you can specify, onto which you can attach a statistical probability?

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Interval Estimation (a.k.a. confidence interval) Is there a range of possible values for that you can specify, onto which you can attach a statistical probability?

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Confidence Interval X - ts X X + ts X _ _ Where t = critical value of t for df = N - 1, two-tailed X = observed value of the sample _

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Tuesday, October 23 Interval estimation. Independent samples t-test for the difference between two means. Matched samples t-test

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Tuesday, October 22 Interval estimation. Independent samples t-test for the difference between two means. Matched samples t-test

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H 0 : 1 - 2 = 0 H 1 : 1 - 2 0

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X boys =53.75 _ X girls =51.16 _ How do we know if the difference between these means, of 53.75 - 51.16 = 2.59, is reliably different from zero?

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X boys =53.75 _ X girls =51.16 _ 95CI: 52.07 boys 55.43 95CI: 49.64 girls 52.68 We could find confidence intervals around each mean...

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H 0 : 1 - 2 = 0 H 1 : 1 - 2 0 But we can directly test this hypothesis...

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H 0 : 1 - 2 = 0 H 1 : 1 - 2 0 To test this hypothesis, you need to know … …the sampling distribution of the difference between means. X 1 -X 2 --

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H 0 : 1 - 2 = 0 H 1 : 1 - 2 0 To test this hypothesis, you need to know … …the sampling distribution of the difference between means. X 1 -X 2 -- …which can be used as the error term in the test statistic.

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X 1 -X 2 = 2 X 1 + 2 X 2 The sampling distribution of the difference between means. This reflects the fact that two independent variances contribute to the variance in the difference between the means. -- --

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X 1 -X 2 = 2 X 1 + 2 X 2 The sampling distribution of the difference between means. This reflects the fact that two independent variances contribute to the variance in the difference between the means. -- -- Your intuition should tell you that the variance in the differences between two means is larger than the variance in either of the means separately.

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The sampling distribution of the difference between means, at n = , would be: z = (X 1 - X 2 ) X 1 -X 2 -- --

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The sampling distribution of the difference between means. Since we don’t know , we must estimate it with the sample statistic s. X 1 -X 2 = 2 1 2 2 n 1 n 2 + --

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The sampling distribution of the difference between means. Rather than using s 2 1 to estimate 2 1 and s 2 2 to estimate 2 2, we pool the two sample estimates to create a more stable estimate of 2 1 and 2 2 by assuming that the variances in the two samples are equal, that is, 2 1 = 2 2. X 1 -X 2 = 2 1 2 2 n 1 n 2 + --

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s X1-X2 = s p 2 s p 2 N 1 N 2 +

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s X1-X2 = s p 2 s p 2 N 1 N 2 +

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s X1-X2 = s p 2 s p 2 N 1 N 2 + s p 2 = SS w SS 1 + SS 2 N-2 =

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Because we are making estimates that vary by degrees of freedom, we use the t-distribution to test the hypothesis. t = (X 1 - X 2 ) - ( 1 - 2 ) s X 1 -X 2 …at (n 1 - 1) + (n 2 - 1) degrees of freedom (or N-2)

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Assumptions X 1 and X 2 are normally distributed. Homogeneity of variance. Samples are randomly drawn from their respective populations. Samples are independent.

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Get district data.

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