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Lecture 36 and 37 - 211 Fall 2009

# Lecture 36 and 37 - 211 Fall 2009 - IV Inference A...

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Unformatted text preview: 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. Handout: U.S. Obesity About 33 percent of men and 35 percent of women were obese …slightly higher than the 31 percent and 33 percent reported in 2003-2004 2003surveys… surveys… However, in generalizing to the U.S. population, researchers calculated a margin of error that swallows up the differences … increases increases not statistically significant. What?!? What?!? 3. Hypothesis Tests for Pop. Proportions Same Basic Concepts: Same 6 Steps Example: U.S. Obesity. 1) Null hypothesis 2) Choose α = 0.05 3) Establish critical value(s): -1.96 1.96 1 3. Hypothesis Tests for Pop. Proportions Complete the Test – Steps 4 – 6. 4) Estimation: zcalc = p2 − π 0 π 0 (1 − π 0 ) n2 5) Compare: 6) Write a Conclusion. 5) Compare: (Two-tail test) -1.96 1.96 5) Compare: (One-tail test) 2 Example: Hypothesis Test for Population Proportion of ResEc 211 Male Students with a Car. Same Basic Concepts: Same 6 Steps Example: U.S. Obesity. 1) Hypothesis: 2) Choose α = 0.10 3) Critical value(s): -1.645 1.645 One view – create confidence intervals. p1 = 0.311 p (1 − p ) p ± zα / 2 ⋅ p2 = 0.333 n n1 ~ n2 ~ 2,200 0.2 0.25 0.3 0.35 0.4 0.45 4. Two-Sample Tests of Proportions We don’t have a “claim” about π – we really have two sample proportions (p1 and p2). Both sample proportions are observations for the random variables p1 and p2. Both sample proportions have standard errors. Both standard errors should be considered in conducting a hypothesis test. 3 4. Two-Sample Test of Proportions. 1) Hypothesis: 2) Choose level of significance. 3) Critical Values: • • Difference – usually a two tail test. Assume the difference is normally distributed. 4. Two-Sample Test of Proportions. 4) Estimation: Two samples ⇒ two sample proportions, p1 and p2 Test Statistic – for the difference (π1 – π2): zcalc = ⋅ ( p2 − p1 ) − (π 2 − π 1 ) π 2 (1 − π 2 ) π 1 (1 − π 1 ) + n2 n1 4. Two Sample Test of Proportions. Test Statistic simplified by our assumptions: zcalc = ( p2 − p1 ) ˆ ˆ ˆ ˆ p (1 − p ) p (1 − p ) + n2 n1 = ( p2 − p1 ) 1 1 ˆ ˆ p (1 − p ) + n2 n1 4 4. Two Sample Test of Proportions. 5) Compare: zcalc vs. zα/2 Final Exam: Topics. Estimation of Population Parameters: Sample mean, sample standard deviation (Ch. 3) Sample proportion (Ch. 11) Sampling Distributions for Sample Mean (Ch 7) Inference for Population Mean Confidence Intervals (Ch. 8) Hypothesis Tests (Ch. 9) Two Sample Tests for Means (Ch. 10, Section 3) Inference for Population Proportions Confidence Intervals (Ch. 11) Hypothesis Tests (Ch. 11) Estimation: Draw Draw a sample – a subset of the population. Estimate Estimate the Population Parameter: 5 Sampling Distributions Sampling Sampling Distribution for Sample Mean. Shape: Normal if Center: Variation: . Sampling Distributions Sampling Sampling Distribution for Sample Proportion. Sample Shape: Center: Variation: Sampling Distributions Areas Areas under the sampling distribution. sampling Given zGiven a z-score or probability, solve for the sample mean: 6 Inference: Point and Interval Estimation. Point Point estimation Estimators Estimators are random variables Confidence Confidence Intervals: Confidence Confidence Intervals are also random. Interval Estimation. Choose Choose Level of Confidence: Prior to estimation, ur exercises in class. Confidence Intervals: Identify the Unknown Population Parameter: 7 Confidence Interval for µ – σx known. Confidence interval for µ - σx not known. Confidence Interval for π . Lower Limit Point Estimate Upper Limit 8 Point and Interval Estimation. Sample Sample Size: You want a certain margin of error given your chosen level of confidence: Solve Solve for n: Hypothesis Tests. We We test “claims” about unknown population unknown parameters. Each Each hypothesis: Null and Alternative. Which Which is the “research question?” Must Must have a specific numeric value – which hypothesis is “equal to” that value? 3 Hypothesis Tests: Identify the Unknown Population Parameter: 9 Six Six steps for any hypothesis test: Step 1: Hypothesis – null and alternative. Step 2: Choose α. Step 3: Determine critical value(s). Draw! critical Step 4: Select a sample, estimate and estimate calculate calculate the test statistic. Step 5: Compare – calculated test statistic to critical value. Place test stat. on picture. Step 6: Write conclusion. 10 10 ...
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