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)

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

Sampling Distribution of a Sample Proportion

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.15 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.15 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.15 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 p ^

• 0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 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?

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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) SEˆ

p =

• p(1 − p)

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 ˆ p ± Z∗ ·

• ˆ

p(1 − ˆ p) 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., np0 and np0(1 − p0) ≥ 10) is the reverse process:

• 1. Convert ˆ

p to a z-score within the theoretical distribution . Zobserved = ˆ p − p0

• p0(1−p0)

n

• 2. Find the relevant area beyond Zobserved 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