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# lecture8note - Psych 100A Lecture 8 Statistical Inference 1...

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1 Psych 100A Winter 2010 Lecture 8: Statistical Inference 1.Interval estimate (recap) 2.Hypothesis testing: tests of significance Example Notations and definitions 3.Test for , known 1 1. Interval estimates How accurate are the point estimates? 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 finds that the sample mean = \$14,400. 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 Solution: Although the exact distribution of incomes for this community is unknown, ) , ( ~ 2 n N x 1. Sampling distribution of mean incomes is normally distributed with mean and variance 2 /n due to central limit theorem. Or, 2. Magnitude of chance errors in sample mean when repeatedly drawing a random sample of size 900 from all 25,000 families is 300 \$ 900 000 , 9 \$ n 3. Chance error in sample mean is called standard error. 3 Solution (cont.): % 95 96 . 1 96 . 1 Pr n x n x 4. Because is normally distributed, the probability that is within 1.96 standard errors of equals: x 5. “1.96” comes from … Table A. 0 25 50 -4 -3 -2 -1 0 1 2 3 4 Units of z % Density 95% 2.5% 2.5% -1.96 1.96 n 96 . 1 n 96 . 1 Units of x 4 Solution (cont.): x equals n x n 96 . 1 96 . 1 96 . 1 96 . 1 6. Algebraic manipulation yields n x n x or n n 96 . 1 96 . 1 7. On repeated random sampling, the true population mean, will be captured by the interval , 95% of the time. n x 96 . 1 5 Solution (cont.): 300 \$ 400 , 14 \$ n x 8. Thus, 300 \$ 96 . 1 400 , 14 \$ equals take" or give " the So, And, we can reasonably expect that the interval (\$13,812, \$14,988) is near the true population mean, of incomes in South Los Angeles. 6

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2 Theorem: confidence interval for the 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 , then is a (1- )100% confidence interval for (CI for ). 7 Comments: CI for , 2 known 1- is called the confidence coefficient and reflects the desired level of accuracy.
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