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## Last Time

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**Last Time**• Central Limit Theorem • Illustrations • How large n? • Normal Approximation to Binomial • Statistical Inference • Estimate unknown parameters • Unbiasedness (centered correctly) • Standard error (measures spread)**Administrative Matters**Midterm II, coming Tuesday, April 6**Administrative Matters**Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators**Administrative Matters**Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas)**Administrative Matters**Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas) • Bring with you: • One 8.5 x 11 inch sheet of paper**Administrative Matters**Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas) • Bring with you: • One 8.5 x 11 inch sheet of paper • With your favorite info (formulas, Excel, etc.)**Administrative Matters**Midterm II, coming Tuesday, April 6 • Numerical answers: • No computers, no calculators • Handwrite Excel formulas (e.g. =9+4^2) • Don’t do arithmetic (e.g. use such formulas) • Bring with you: • One 8.5 x 11 inch sheet of paper • With your favorite info (formulas, Excel, etc.) • Course in Concepts, not Memorization**Administrative Matters**Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10**Administrative Matters**Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10 • Note: due Thursday, April 2**Administrative Matters**Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10 • Note: due Thursday, April 2 • Will ask grader to return Mon. April 5 • Can pickup in my office (Hanes 352)**Administrative Matters**Midterm II, coming Tuesday, April 6 • Material Covered: HW 6 – HW 10 • Note: due Thursday, April 2 • Will ask grader to return Mon. April 5 • Can pickup in my office (Hanes 352) • So today’s HW not included**Administrative Matters**Extra Office Hours before Midterm II Monday, Apr. 23 8:00 – 10:00 Monday, Apr. 23 11:00 – 2:00 Tuesday, Apr. 24 8:00 – 10:00 Tuesday, Apr. 24 1:00 – 2:00 (usual office hours)**Study Suggestions**• Work an Old Exam • On Blackboard • Course Information Section**Study Suggestions**• Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions**Study Suggestions**• Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions • Rework HW problems**Study Suggestions**• Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions • Rework HW problems • Print Assignment sheets • Choose problems in “random” order**Study Suggestions**• Work an Old Exam • On Blackboard • Course Information Section • Afterwards, check against given solutions • Rework HW problems • Print Assignment sheets • Choose problems in “random” order • Rework (don’t just “look over”)**Reading In Textbook**Approximate Reading for Today’s Material: Pages 356-369, 487-497 Approximate Reading for Next Class: Pages 498-501, 418-422, 372-390**Law of Averages**Case 2: any random sample CAN SHOW, for n “large” is “roughly” Terminology: • “Law of Averages, Part 2” • “Central Limit Theorem” (widely used name)**Central Limit Theorem**Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 25 (seems very mound shaped?)**Extreme Case of CLT**Consequences: roughly roughly Terminology: Called The Normal Approximation to the Binomial**Normal Approx. to Binomial**How large n? • Bigger is better • Could use “n ≥ 30” rule from above Law of Averages • But clearly depends on p • Textbook Rule: OK when {np ≥ 10 & n(1-p) ≥ 10}**Statistical Inference**Idea: Develop formal framework for handling unknowns p & μ e.g. 1: Political Polls e.g. 2a: Population Modeling e.g. 2b: Measurement Error**Statistical Inference**A parameter is a numerical feature of population, not sample An estimate of a parameter is some function of data (hopefully close to parameter)**Statistical Inference**Standard Error: for an unbiased estimator, standard error is standard deviation Notes: • For SE of , since don’t know p, use sensible estimate • For SE of , use sensible estimate**Statistical Inference**Another view: Form conclusions by**Statistical Inference**Another view: Form conclusions by quantifying uncertainty**Statistical Inference**Another view: Form conclusions by quantifying uncertainty (will study several approaches, first is…)**Confidence Intervals**Background:**Confidence Intervals**Background: The sample mean, , is an “estimate” of the population mean,**Confidence Intervals**Background: The sample mean, , is an “estimate” of the population mean, How accurate?**Confidence Intervals**Background: The sample mean, , is an “estimate” of the population mean, How accurate? (there is “variability”, how much?)**Confidence Intervals**Idea: Since a point estimate (e.g. or )**Confidence Intervals**Idea: Since a point estimate is never exactly right (in particular )**Confidence Intervals**Idea: Since a point estimate is never exactly right give a reasonable range of likely values (range also gives feeling for accuracy of estimation)**Confidence Intervals**Idea: Since a point estimate is never exactly right give a reasonable range of likely values (range also gives feeling for accuracy of estimation)**Confidence Intervals**E.g.**Confidence Intervals**E.g. with σ known**Confidence Intervals**E.g. with σ known Think: measurement error**Confidence Intervals**E.g. with σ known Think: measurement error Each measurement is Normal**Confidence Intervals**E.g. with σ known Think: measurement error Each measurement is Normal Known accuracy (maybe)**Confidence Intervals**E.g. with σ known Think: population modeling**Confidence Intervals**E.g. with σ known Think: population modeling Normal population**Confidence Intervals**E.g. with σ known Think: population modeling Normal population Known s.d. (a stretch, really need to improve)**Confidence Intervals**E.g. with σ known Recall the Sampling Distribution:**Confidence Intervals**E.g. with σ known Recall the Sampling Distribution: (recall have this even when data not normal, by Central Limit Theorem)**Confidence Intervals**E.g. with σ known Recall the Sampling Distribution: Use to analyze variation**Confidence Intervals**Understand error as: (normal density quantifies randomness in )**Confidence Intervals**Understand error as: (distribution centered at μ)**Confidence Intervals**Understand error as: (spread: s.d. = )