ch2_3 - Review of Probability and Statistics (SW Chapters...

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2&3-14 Review of Probability and Statistics (SW Chapters 2,3) Empirical problem: Class size and educational output Policy question: What is the effect of reducing class size by one student per class? by 8 students/class? What is the right output measure (“dependent variable”)? ± parent satisfaction ± student personal development ± future adult welfare and/or earnings ± performance on standardized tests
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2&3-15 What do data say about the class size/test score relation? The California Test Score Data Set All California elementary school districts ( n = 420) Variables: ± 5 th grade test scores (Stanford-9 achievement test, combined math and reading), district average ± Student-teacher ratio (STR) = no. of students in the district divided by no. full-time equivalent teachers
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2&3-16 Initial look at the data:
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2&3-17 Do districts with smaller classes have higher test scores?
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2&3-18 How can we get some numerical evidence on whether districts with low STRs have higher test scores? 1. Compare average test scores in districts with low STRs to those with high STRs (“ estimation ”) 2. Test the hypothesis that the mean test scores in the two types of districts are the same, against the alternative hypothesis that they differ (“ hypothesis testing ”) 3. Estimate an interval for the difference in the mean test scores, high v. low STR districts (“ confidence interval ”) Initial data analysis: Compare districts with “small” (STR < 20) and “large” (STR 20) class sizes:
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2&3-19 Class Size Average score ( Y ) Standard deviation ( s Y ) n Small 657.4 19.4 238 Large 650.0 17.9 182 1. Estimation of = difference between group means 2. Test the hypothesis that = 0 3. Construct a confidence interval for
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2&3-20 1. Estimation small large YY = 657.4 – 650.0 = 7.4 where small small 1 small 1 n i i n = = and large large 1 large 1 n i i n = = Is this a large difference in a real-world sense? Standard deviation across districts = 19.1 Difference between 60 th and 75 th percentiles of test score distribution is 667.6 – 659.4 = 8.2 This is a big enough difference to be important for school reform discussions, for parents, or for a school committee
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2&3-21 2. Hypothesis testing Difference-in-means test: compute the t -statistic, 22 () sl ss s l nn YY t SE Y Y −− == + (remember this?) where SE ( s Y l Y ) is the “standard error” of s Y l Y ; the subscripts s and l refer to “small” and “large” STR districts; and 1 1 1 s n s is i s sY Y n = =− (etc.)
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2&3-22 Compute the difference-of-means t -statistic: Size Y s Y n small 657.4 19.4 238 large 650.0 17.9 182 22 2 2 19.4 17.9 238 182 657.4 650.0 7.4 1.83 sl ss nn YY t == = ++ = 4.05 |t| > 1.96, so reject (at the 5% significance level) the null hypothesis that the two means are the same.
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2&3-23 3. Confidence interval A 95% confidence interval for the difference between the means is, ( s Y l Y ) ± 1.96 × SE ( s Y l Y ) = 7 . 4 ± 1.96 × 1.83 = (3.8, 11.0) Two equivalent statements: 1. The 95% confidence interval for doesn’t include 0; 2. The hypothesis that = 0 is rejected at the 5% level.
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2&3-24 This should all be familiar. But: 1.
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This note was uploaded on 09/09/2008 for the course ECON 3412 taught by Professor Vonwachter during the Fall '08 term at Columbia.

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ch2_3 - Review of Probability and Statistics (SW Chapters...

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