Mon Nov 21, 2016

Wrap Up

  1. In general: What 3 components exist for all hypothesis tests
  2. Specifically: How the 3 components vary by setting

There is Only One Test

Hypothesis Testing in General

The how of these 3 points may change, but the what doesn't:

  1. Identify the test statistic
  2. Construct the null distribution of the test statistic based on \(H_0\)
  3. Compare the observed test statistic to the null distribution to compute the p-value

Specifics 1: Test Statistic

We can do hypothesis testing for different scenarios

Type Population Parameter Test Statistic
One-Sample Mean \(\mu\) Sample Mean \(\overline{x}\)
One-Sample Proportion \(p\) Sample Proportion \(\widehat{p}\)
Two-Sample Diff of Means \(\mu_1 - \mu_2\) \(\overline{x}_1 - \overline{x}_2\)
Two-Sample Diff of Proportions \(p_1 - p_2\) \(\widehat{p}_1 - \widehat{p}_2\)

Specifics 2: Null Distribution

We can construct the null distribution of the test statistic either

  • Via computation i.e. simulation/sampling
  • Ex: We used the Permutation Test i.e. we permuted things
  • Via mathematics i.e. analytically
  • Ex: In MATH 310 Probability via the Central Limit Theorem: \(\overline{X}\) is Normally distributed as \(n \rightarrow \infty\)
  • Note: the null distribution isn't always bell shaped! i.e. not always Normal

Specifics 3: p-Value

Depending on the alternative hypothesis \(H_A\), we have either

  1. Two-sided p-values. Ex:
    • \(H_A: \mu_{1} - \mu_{2} \neq 0\)
  2. One-sided p-values. Ex:
    • \(H_A: \mu_{1} - \mu_{2} < 0\)
    • \(H_A: \mu_{1} - \mu_{2} > 0\)

Learning Check from Lec25

So the LC from Lec25 involving evens vs odds, we had

  • \(H_0: \mu_{odd} - \mu_{even} = 0\) vs
  • \(H_A: \mu_{odd} - \mu_{even} \neq 0\)

This is a two-sided permutation test for differences in means.

Moral of the Story

If you forget what hypothesis testing and/or p-values are remember: