Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test STAT 113 Normal-Based Inference for Proportions Colin Reimer Dawson Oberlin College November 11-12, 2019 1 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Outline Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 2 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Limits of Normal Approximation So Far • So far we have still needed to do simulation to calculate the standard error • We can avoid that with some more theory 3 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Cases to Address We will need standard errors to do CIs and tests for the following parameters: 1. Single Proportion (now) 2. Single Mean (tomorrow) 3. Difference of Proportions (tomorrow?) 4. Difference of Means (tomorrow?) 5. Mean of Differences (Thursday?) 4 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Analytic Approximations of Sampling Distributions Param. Stat. Theory SE Distribution df � p (1 − p ) p p ˆ N (0 , 1) – n � s 2 n − 1 µ x ¯ t n � SE 2 p A + SE 2 p A − p B p A − ˆ ˆ p B N (0 , 1) – ˆ ˆ p B � SE 2 p A + SE 2 µ A − µ B x A − ¯ min( n A , n B ) − 1 ¯ x B t ˆ p B ˆ � s 2 diff n diff − 1 µ diff x diff ¯ t n diff � 1 − r 2 ρ r t n − 2 n − 2 Observed Statistic ± Standardized Quantile × � CI : SE Observed Statistic − Null Param . Sandardized Test Statistic : � SE 5 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Outline Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 6 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Sampling Distribution of a Sample Proportion 0.4 0.4 0.4 ● ● ● ● 0.2 0.2 ● 0.2 ● ● ● ● ● ● ● ● ● ● 0.0 0.0 0.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ^ ^ ^ p p p ● ● ● ● 0.15 0.15 0.15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.00 0.00 ● ● ● ● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ^ ^ ^ p p p 0.04 0.04 0.04 ● ● ● ● ● ● 0.02 ● 0.02 ● 0.02 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.00 ● ● 0.00 ● ● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ^ ^ ^ p p p Columns: values of p (left: 0.1, middle: 0.5; right: 0.9) Rows: values of n (top: 10, middle: 50; bottom: 1000) • Larger samples and more population homogeneity make SE go down 7 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Sampling Distribution of ˆ p • Condition: Use a Normal approximation with at least 10 expected cases of each outcome: np ≥ 10 n (1 − p ) ≥ 10 • Mean: p • Standard deviation (standard error): � p (1 − p ) SE ˆ p = n 8 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Outline Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 9 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test CI Summary: Single Proportion 0. Check whether conditions for distribution approximation hold: n ˆ p > = 10 and n (1 − ˆ p ) ≥ 10 1. If so, find the mean and SD of the theoretical distribution to replace the bootstrap distribution • Mean = Sample Statistic = ˆ p � p (1 − ˆ ˆ p ) • SD = Standard Error = n 2. Use the confidence level and a Standard Normal to get z -scores of the endpoints • 95%: z = ± 1 . 96 ( ≈ 2 ) • 99%: z = ± 2 . 58 • 90%: z = ± 1 . 64 3. Convert z scores to endpoints on the original scale using the mean and standard deviation found in step 1. Endpoint = Sample Statistic + z · SE 10 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Example: Kissing Right Most people are right-handed, and even the right eye is dominant for most people. Developmental biologists have suggested that late-stage human embryos tend to turn their heads to the right. In a study reported in Nature (2003), German bio-psychologist Onur Güntürkün studied kissing couples in public places such as airports, train stations, beaches, and parks. They observed 124 couples, age 13-70 years. For each kissing couple observed, the researchers noted whether the couple leaned their heads to the right or to the left. Of the 124 couples, 80 leaned right. Let’s find a 95% confidence interval for p , the proportion of all couples in the target population who would lean right. 11 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Kissing Right: Descriptive Stats and Conditions • Population Parameter (what we want to estimate): p , proportion of all couples who would lean right • Sample Statistic (what we have): ˆ p , proportion of couples in this study who leaned right p = 80 / 124 = 0 . 645 ˆ • Conditions: n ˆ p ≥ 10 and n (1 − ˆ p ) ≥ 10 n ˆ p = 124 × 0 . 645 = 80 n (1 − ˆ p ) = 124 × 0 . 355 = 44 so we are okay to use the Normal distribution approximation. 12 / 21
Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test Kissing Right: Standard Error � p (1 − p ) • Standard Error for a proportion: SE ˆ p = n � p (1 − ˆ ˆ p ) ˆ • Estimate with SE ˆ p = n • We have n = 124 and ˆ p = 0 . 645 : � p (1 − ˆ ˆ p ) ˆ SE ˆ p = n � 0 . 645 × 0 . 355 = 124 = 0 . 043 13 / 21
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