Lecture 34 - 211 Fall 2009

Lecture 34 - 211 Fall 2009 - IV. Inference. A. Confidence...

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Unformatted text preview: IV. Inference. A. Confidence Intervals for Population Mean. (Chapter 8) 8) B. Hypothesis Tests of Population Mean (Chapter 9) 1. Basics of Hypothesis Testing 2. Errors, Terms and Conclusions 3. Hypothesis tests: Z-test or t-test (is σX known?) Zt4. Prob Values (P-values) (P5. Type I and Type II Error Probabilities and Power of the Test C. Hypothesis Test for Two Pop. Means (Ch. 10) 10) 5. Type I Error, Type II Error and Power of the Test Type I Error – Type II Error – The Power of the test – Draw each of these! P(Type I Error) = α if H0 is true P(Type II Error) = β if H0 is false Trade-Offs - If H0 false: decreasing α, increases P(Type II Error) = β decreasing α, decreases Power ( 1 - β ) 1 • Illustrate – hypothesis test and level of significance. • Assume H0 is wrong – Illustrate Type II Error. α = 0.05 700 0 1.645 hours Z • Suppose we worry about a Type I Error – reduce α • Power – area to the right of the critical value. α = 0.01 hours Z 700 0 1.96 • Suppose we increase sample size. α = 0.01 hours Z 700 0 1.96 2 C. Hypothesis Test for Two Pop. Means (Sections 10.1 and 10.3) Example: Does regular PRS participation help to improve students’ learning/performances? 1) Hypothesis – Null and Alternative. Null Hypothesis: Alternative Hypothesis: Estimation: Estimation: Estimate Estimate both μR and μN: Sample Sample – from both populations. populations. n Regulars NonNon-users 176 117 X 78.24 54.97 s 14.88 30.44 Random Random variable: Standard Standard error: 2) Choose α: α = 0.01 3) Critical Value: t(0.01,??) = ????? Degrees of Freedom for two sample test. In In general, when σ1 and σ2 are not known: df =Δ= 2 ⎡ ⎛ s12 ⎞ ⎛ s2 ⎞ ⎤ ⎢ ⎜ n1 ⎟ + ⎜ n2 ⎟ ⎥ ⎠ ⎝ ⎠⎦ ⎣⎝ 2 2 2 ⎛ s12 ⎞ ⎛ s2 ⎞ ⎜ n⎟ ⎜ n⎟ 1⎠ 2⎠ ⎝ +⎝ n1 − 1 n2 − 1 2 If: If: you find s1 and s2 are close in value and n1 and n2 are close in value, then you can use: df = n1+ n2 – 2. df 3 Standardized Test: Once Once the hypothesis test is set-up: set4) Estimate: Samples of regular PRS users versus “non-users” in ResEc 211 (Fall 2008). n Regulars NonNon-users tcalc = 2 R 176 117 XR − XN X 78.24 54.97 s 14.88 30.44 2 ⎛s ⎞ ⎛ sR ⎞ ⎜ n⎟+⎜ n⎟ R⎠ R⎠ ⎝ ⎝ 5) Compare – complete the Standardized t-test: Standardized t- 6) Conclusion? 4 IV. Inference. A. Confidence Intervals for Population Mean. (Ch. 8) B. Hypothesis Tests for Population Mean (Ch. 9) C. Hypothesis Test for Two Population Means (Ch. 10) D. Inference for Pop. Proportions (Ch. 11) 1. The Sampling Distribution for p 2. Confidence intervals for π 3. Hypothesis tests for π 4. Two Sample tests for proportions. C. Inference for Population Proportions 1. Sampling Distributions for Proportions Population proportion – a parameter: ResEc ResEc 211 Survey Results: Did Did you bring with you a … Item TV Stereo 2004 0.614 0.468 2006 0.671 0.389 0.209 0.783 2007 0.619 0.314 0.181 0.808 2008 0.618 0.270 0.086 0.900 2009 Desktop 0.475 Laptop 0.479 5 1. Sampling Distributions for Proportions Sample proportion – an estimate of the population proportion: 1. Sampling Distributions for Proportions Sampling Distribution for Sample Proportion Three Characteristics of Interest: • Shape: When is the S.D. normal? 1. Sampling Distributions for Proportions • Center: • Variation: Sampling Error = Standard Error – 6 ...
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