6_1_Handout - 6.1 2003 those who took the SAT SAT verbal...

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6.1 Estimating with Confidence 2003- those who took the SAT SAT verbal: mean= 507, sd= 111 SAT math: mean= 519, sd= 115 give test to SRS of 500 California students: x = 461 (math) What can you say about the mean score μ in population of 385,000? x is an unbiased estimator of μ (μ x = μ) • but how reliable is this estimate? (how variable is the statistic?) recall x is approximately N (μ, σ/ n ) when n is large suppose σ = 100, then σ x = 4.5 (unrealistic to assume we would know σ- CH 7) x has normal distribution centered at unknown population mean μ with σ x = 4.5 Figure 6.2 •the 68-95-99.7 rule says that the probability is about 0.95 that x will be within 9 points (two standard deviations of x ) of the population mean μ •to say that x lies within 9 points of μ is the same as saying that μ is within 9 points of x •so 95% of all samples will capture the true μ in the interval from x - 9 to x + 9 our sample gave x = 461 we say that we are 95% confident that the unknown mean score lies between 452 and 470 There are two possibilities: 1) The interval between 452 and 470 contains the true μ 2) Our SRS was one of the few samples for which x is not within 9 points of the true μ. Only 5% of samples give such inaccurate results.
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Confidence intervals the interval of numbers between the values x ± 9 is called a 95% confidence interval for μ confidence interval = estimate ± margin of error estimate= x
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