Inference on Single Mean

This is called the most efficient estimator efficient

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Unformatted text preview: cient Sampling Distributions for Mean and Median mean median -8 -6 -4 -2 0 2 4 6 8 For normal populations, the sample mean is the most efficient G. Baker, Department of Statistics G. estimator of µ. University of South Carolina; Slide 39 University 39 Interval Estimate of the Mean Yn − µ Z= follows a standard normal distribution σ/ n Y −µ P (−1.96 < < +1.96) = 0.95 σ/ n P( µ = Y ± 1.96 σ n (with a little algebra) ) = 0.95 So we say that we are 95% confident that µ is in the interval Y ± 1 . 96 σ n What assumptions have we made? G. Baker, Department of Statistics G. University of South Carolina; Slide 40 University 40 Interval Estimate of the Mean Standard Normal 0.95 .025 -4 -3 -2 -1.96 -1 0 .025 1 2 1.96 3 4 Z G. Baker, Department of Statistics G. University of South Carolina; Slide 41 University 41 Interval Estimate of the Mean Let’s go from 95% confidence to the Let go general case. general The symbol zα is the z-value that has an value area of α to the right of it. P(− zα / 2 Y −µ < < + zα / 2 ) = (1 − α ) σ/ n P ( µ = Y ± zα / 2 σ n ) = (1 − α ) G. Baker, Department of Statistics G. University of South Carolina; Slide 42 University 42 Interval Estimate of the Mean Standard Normal 1-α α/2 -4 -3 -Zα/2 -2 -1 0 α/2 1 +Zα/22 3 4 (1 – α) 100% Confidence Interval G. Baker, Department of Statistics G. University of South Carolina; Slide 43 University 43 What Does (1 – α) 100% Confidence Mean? What Sampling Distribution of the y n( µ , σ / n) y y Z y 8 7 S mle ap x y 6 5 y 4 3 2 y y x y y 1 xy x (1-α)100% Confidence Intervals 0 µ G. Baker, Department of Statistics G. University of South Carolina; Slide 44 University 44 If Z0.05 = 1.645, we are _____% 1.645, confident that the mean is between y ± 1.645 σ n A.99% B.95% C.90% D.85% G. Baker, Department of Statistics G. University of South Carolina; Slide 45 University 45 Which z-value would you use to value calculate a 99% confidence interval on a mean? interval Z0.10 = 1.282 B. Z0.01 = 2.326 C. Z0.005 = 2.576 D. Z0.0005 = 3.291 A. G. Baker, Department of Statistics G. University of South Carolina; Slide 46 University 46 Plastic Injection Molding Process A plastic injection molding process for a part that has a critical width dimension historically follows a normal distribution with a standard deviation of 8. with Periodically, clogs from one of the feeder lines causes the mean width to change. As a result, the operator periodically takes random samples of size 4. random G. Baker, Department of Statistics G. University of South Carolina; Slide 47 University 47 Plastic Injection Molding A recent sample of four yielded a sample mean of 101.4. mean Construct a 95% confidence interval for the true mean width. the Construct a 99% confidence for the true mean width. mean G. Baker, Department of Statistics G. University of South Carolina; Slide 48 University 48 When going from a 95% confidence When interval to a 99% confidence interval, the width of the interval will the Increase. B. Decrease. C. Remain t...
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