ch19 - STAT 2023 - Holbrook Confidence Intervals for...

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Unformatted text preview: STAT 2023 - Holbrook Confidence Intervals for Proportions Chapter 19 19 - 1 STAT 2023 - Holbrook Categorical Data 19 - 2 STAT 2023 - Holbrook Categorical Data 1. Categorical Random Variables Yield Categorical Responses That Classify Responses e.g., Gender (Male, Female) 2. Measurement Reflects # in Category 3. Examples 19 - 3 Have you ever smoked marijuana? Have Do you have a tattoo? Have you pierced something other than Have your ears? your STAT 2023 - Holbrook Proportions 1. Involve Categorical Variables 2. Fraction or % of population or sample in a Fraction category category 3. If Two Categorical Outcomes, Binomial If Distribution Distribution 19 - 4 Possess or Don’t Possess Characteristic STAT 2023 - Holbrook Proportions 1. Involve Categorical Variables 2. Fraction or % of population or sample in a Fraction category category 3. If Two Categorical Outcomes, Binomial If Distribution Distribution Possess or Don’t Possess Characteristic ^ 4. Sample Proportion (p) 19 - 5 = x = number of successes p n sample size Thinking Challenge STAT 2023 - Holbrook Suppose you’re Suppose interested in the proportion of students at SFCC (the population) who have ever smoked marijuana. How would you find out? out? 19 - 6 Statistical Methods STAT 2023 - Holbrook Statistical Methods Descriptive Statistics Inferential Statistics Estimation 19 - 7 Hypothesis Testing Estimation Process STAT 2023 - Holbrook Population Proportion, p, is unknown 19 - 8 Estimation Process STAT 2023 - Holbrook Population Proportion, p, is unknown Sample 19 - 9 Random Sample Prop. p = .60 ^ Estimation Process STAT 2023 - Holbrook Population Proportion, p, is unknown Sample 19 - 10 10 Random Sample Prop. p = .60 ^ I am 95% confident that p is between . 50 & .70. STAT 2023 - Holbrook Unknown Population Parameters Are Estimated Estimate Population Parameter... µ Mean with Sample Statistic y p ^ p p1 – p2 ^ ^ p1 – p2 Diff. in means µ 1 - µ 2 Diff. y1 -y2 Proportion Diff. in prop. 19 - 11 11 Estimation Methods STAT 2023 - Holbrook Estimation Point Estimation 19 - 12 12 Interval Estimation STAT 2023 - Holbrook Point Estimation 19 - 13 13 Estimation Methods STAT 2023 - Holbrook Estimation Point Estimation 19 - 14 14 Interval Estimation Point Estimation STAT 2023 - Holbrook 1. Provides Single Value Based on Observations from 1 Sample 2. Gives No Information about How Close Gives Value Is to the Unknown Population Parameter Parameter ^ 3. Example: Sample Proportion p = .60 Is the Point Estimate of Unknown Population Proportion Population 19 - 15 15 STAT 2023 - Holbrook Interval Estimation 19 - 16 16 Estimation Methods STAT 2023 - Holbrook Estimation Point Estimation 19 - 17 17 Interval Estimation Interval Estimation STAT 2023 - Holbrook 1. Provides Range of Values Provides Based on Observations from 1 Sample 2. Gives Information about Closeness to Unknown Gives Population Parameter Population Stated in terms of Probability Knowing Exact Closeness Requires Knowing Unknown Knowing Population Parameter Population 3. Example: Unknown Population Proportion Lies Example: Between .50 & .70 with 95% Confidence Between 19 - 18 18 STAT 2023 - Holbrook Key Elements of Interval Estimation Sample statistic Sample (point estimate) (point 19 - 19 19 STAT 2023 - Holbrook Key Elements of Interval Estimation Confidence Confidence interval interval Confidence Confidence limit (lower) limit 19 - 20 20 Sample statistic Sample (point estimate) (point Confidence Confidence limit (upper) limit STAT 2023 - Holbrook Key Elements of Interval Estimation A probability that the population parameter probability falls somewhere within the interval. falls Confidence Confidence interval interval Confidence Confidence limit (lower) limit 19 - 21 21 Sample statistic Sample (point estimate) (point Confidence Confidence limit (upper) limit STAT 2023 - Holbrook JUMPING AHEAD! Derivation of Confidence Interval Formula for the Mean– Chapter 23 (same concept applies to confidence interval for proportions) 19 - 22 22 Confidence Limits for Population Mean STAT 2023 - Holbrook Parameter = Statistic ± Error Error = Y − µ or Y + µ (3) Y − µ Error Z= = σy σy (4) 19 - 23 23 µ = Y ± Error (2) © 1984-1994 T/Maker Co. (1) Error = Zσy (5) µ = Y ± Zσy Intervals & Confidence Level STAT 2023 - Holbrook Sampling Sampling Distribution α /2 of Mean of 1- α σ x_ Y α /2 µ Yx = µ Y X (1 - α ) % of of intervals contain µ . Intervals Intervals extend from Y- Zσ Y to Y + Zσ Y α % do not. do Large number of intervals 19 - 24 24 _ Confidence Level STAT 2023 - Holbrook 1. Probability that the Unknown Probability Population Parameter Falls Within Interval Interval 2. Denoted (1 - α) % Denoted α) α Is Probability That Parameter Is Not Not Within Interval Within 3. Typical Values Are 99%, 95%, 90% 19 - 25 25 Z­Values STAT 2023 - Holbrook 1. 2. 3. 4. For a 99% C.I. Z=2.576 For a 95% C.I. Z=1.96 For a 90% C.I. Z=1.645 See Appendix D, Table T Notice confidence levels at bottom of the Notice page. page. 19 - 26 26 Appendix E, Page A-53 (First Edition) STAT 2023 - Holbrook Factors Affecting Interval Width 1. Data Dispersion Measured by σ Measured Intervals Extend from Y - Zσ Y toY + Zσ Y 2. Sample Size — σY = σ / √n 3. Level of Confidence Level (1 - α) 19 - 27 27 Affects Z © 1984-1994 T/Maker Co. STAT 2023 - Holbrook Note: Page 493; we will only concern ourselves with checking the “success/failure” condition. Page 358­359 (First Edition). 19 - 28 28 STAT 2023 - Holbrook Confidence Interval Estimate of a Proportion (back to Chapter 19) 19 - 29 29 Confidence Interval Proportion STAT 2023 - Holbrook 1. Assumptions Two Categorical Outcomes Normal Approximation Can Be Used (Page Normal 439.) 439.) 19 - 30 30 ˆ ˆ np > 10 and n (1 − p) > 10 Confidence Interval Proportion STAT 2023 - Holbrook 1. Assumptions Two Categorical Outcomes Normal Approximation Can Be Used (Page Normal 439.) 439.) 2. ˆ ˆ np > 10 and n (1 − p) > 10 Confidence Interval Estimate ˆ ˆ ˆ ˆ p ⋅ (1 − p) p ⋅ (1 − p) ˆ ˆ p−z ⋅ ≤p≤p+z ⋅ n n 19 - 31 31 STAT 2023 - Holbrook Estimation Example Proportion A random sample of 400 graduates random showed 32 went to grad school. Set up a 95% confidence interval estimate for p. 19 - 32 32 STAT 2023 - Holbrook Estimation Example Proportion A random sample of 400 graduates random showed 32 went to grad school. Set up a 95% confidence interval estimate for p. (Checking the assumption that the sample size is large (Checking enough.) enough.) ˆ np > 10 or 400(0.08) = 32 > 10 ˆ and n (1 − p) > 10 or 400(1 − 0.08) = 368 > 10 So the sample size is large enough. 19 - 33 33 Estimation Example Proportion STAT 2023 - Holbrook A random sample of 400 graduates random showed 32 went to grad school. Set up a 95% confidence interval estimate for p. ˆ ˆ ˆ ˆ p ⋅ (1 − p) p ⋅ (1 − p) ˆ ˆ p − Z⋅ ≤p≤p+Z ⋅ n n .08 ⋅ (1 − .08) .08 ⋅ (1 − .08) .08 − 1.96 ⋅ ≤ p ≤ .08 + 1.96 ⋅ 400 400 19 - 34 34 .053 ≤ p ≤ .107 Interpretation STAT 2023 - Holbrook I’m 95% confident that the population I’m proportion (p) of graduates who went to grad school is between .053 and .107. grad 19 - 35 35 STAT 2023 - Holbrook Confidence Interval Solution 1. Using the TI-83 Hit “STAT” scroll to “TESTS” then “A”, hit ”, “ENTER” “1-PropZInt“ Enter x and n. For example: x=32 and n=400 Choose a “.95” “C-Level” Select “Calculate” and hit “ENTER” 19 - 36 36 Thinking Challenge STAT 2023 - Holbrook You’re a production You’re manager for a newspaper. You want to find the % defective. Of 200 200 newspapers, 35 had 35 defects. What is the 90% confidence interval 90% estimate of the population proportion defective? proportion 19 - 37 37 STAT 2023 - Holbrook Confidence Interval Checking the Assumption* ˆ np > 10 or 200(0.175) = 35 > 10 ˆ and n (1 − p) > 10 or 200(1 − 0.175) = 165 > 10 So the sample size is large enough. 19 - 38 38 STAT 2023 - Holbrook Confidence Interval Solution* ˆ ˆ ˆ ˆ p ⋅ (1 − p) p ⋅ (1 − p) ˆ ˆ p−z ⋅ ≤ p≤ p+z ⋅ n n .175 ⋅ (.825) .175 ⋅ (.825) .175 − 1.645 ⋅ ≤ p ≤ .175 + 1.645 ⋅ 200 200 .1308 ≤ p ≤ .2192 19 - 39 39 Interpretation STAT 2023 - Holbrook I’m 90% confident that the population I’m proportion (p) of defective newspapers is between .1308 and .2192. between 19 - 40 40 End of Chapter Any blank slides that follow are blank intentionally. ...
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This note was uploaded on 12/07/2011 for the course STA 2023 taught by Professor Staff during the Fall '11 term at Santa Fe College.

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