STAT 113 Analytic Inference for a Single Proportion Colin Reimer - PowerPoint PPT Presentation

STAT 113 Analytic Inference for a Single Proportion Colin Reimer Dawson Oberlin College 7-10 April 2017

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion Limits of Normal Approximation So Far • We have still needed to do all that randomization / resampling to calculate the standard error. • We can avoid that with some more theory. 4 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion Cases to Address We will need standard errors to do CIs and tests for the following parameters: 1. Single Proportion (now) 2. Single Mean (Wednesday) 3. Difference of Proportions (Friday) 4. Difference of Means (Friday) 5. Mean of Differences (new! next week) 5 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion 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) 6 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion Things Affecting the Standard Error for ˆ p 1. Sample Size ( n ) • Increasing n makes the standard error go 2. Population Proportion ( p ) • What values of p make SE larger? 7 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion Distribution of ˆ p When the population proportion is p and the samples are of size n , the sampling distribution of ˆ p has mean p and standard deviation (standard error) � p (1 − p ) p = SE ˆ n It is also approximately Normal, when samples are large enough, and p isn’t too extreme. Rough rule: at least 10 (expected) cases each with “positive” and “negative” outcome. 8 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion CI Summary: Single Proportion To compute a confidence interval for a proportion when the bootstrap distribution for ˆ p is approximately Normal (i.e., counts for both outcomes ≥ 10 ), use � p (1 − ˆ ˆ p ) p ± Z ∗ · ˆ n where Z ∗ is the Z -score of the endpoint appropriate for the confidence level, computed from a standard normal ( N (0 , 1) ). 10 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion Example: Kissing Right CI Demo: Three methods 11 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion P -values for a sample proportion from a Standard Normal Computing P -values when the null sampling distribution is approximately Normal (i.e., np 0 and np 0 (1 − p 0 ) ≥ 10 ) is the reverse process: 1. Convert ˆ p to a z -score within the theoretical distribution . p − p 0 ˆ Z observed = � p 0 (1 − p 0 ) n 2. Find the relevant area beyond Z observed using a Standard Normal 13 / 14

Outline Theoretical Approximation of SE CI for Sample Proportion Test for a Proportion Example: Kissing Right Hypothesis Test Demo: Three methods 14 / 14