lecture8 - Psych 100A Lecture 8: Statistical Inference 1....

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Psych 100A Winter 2010 Lecture 8: Statistical Inference 1. Interval estimate (recap) 2. Hypothesis testing: tests of gnificance significance Example Notations and definitions 3. Test for , known 1
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1. Interval estimates How accurate are the point estimates? xample: uppose a city manager in Los Angeles Example: Suppose a city manager in Los Angeles wants to know the mean income ( ) of 25,000 families living in South Los Angeles. She interviews 900 (n) families “at random” and finds that the sample mean = $14,400 . Based on past study, she knows that the population SD ( ) of incomes equals $9,000. Problem: Put a “give or take” number on the point estimator (ie, the sample mean). x ? 2
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Solution: Although the exact distribution of comes for this community is unknown incomes for this community is unknown, 1. Sampling distribution of mean incomes is ormally distributed ith mean nd 2 normally distributed with mean and variance 2 /n due to central limit theorem. Or, ) , ( ~ n N x 2. Magnitude of chance errors in sample mean hen repeatedly drawing a random sample of when repeatedly drawing a random sample of size 900 from all 25,000 families is 00 300 $ 900 000 , 9 $ hance error ample mean called 3. Chance error in sample mean is called standard error . 3
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Solution (cont.): ecause ormally distributed the x 4. Because is normally distributed, the probability that is within 1.96 standard errors of equals: % 95 96 . 1 96 . 1 Pr n 5. “1.96” comes from … Table A. 50 ity -1.96 1.96 25 % Dens 95% 2.5% 2.5% 0 -4 -3 -2 -1 0 1 2 3 4 Units of z 96 . 1 96 . 1 Units of 4
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Solution (cont.): 6. Algebraic manipulation yields equals n x 96 . 1 96 . 1 96 . 1 96 . 1 or 96 . 1 96 . 1 7. On repeated random sampling, the true population mean, will be captured by the interval , 95% of the time.   96 . 1 5
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olution (cont ): Solution (cont.): 8. Thus, 00 400 , 14 $ x 300 $ equals take" or give " the So, n   300 $ 96 . 1 400 , 14 $ And, we can reasonably expect that the interval ($13,812, $14,988) is near the true population mean, of incomes in South Los ngeles. Angeles. 6
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population mean, 2 known If x 1 , x 2 , …, x n is a random sample from a normal population or a large random sample from any population with unknown mean and known variance 2 , then   n z x 2 is a (1- )100% confidence interval for (CI for ). 7
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lecture8 - Psych 100A Lecture 8: Statistical Inference 1....

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