Section 22 Hypothesis Testing - Two Samples: Concept

22.1 Two Samples, Unknown Variance

Inference for a difference in means of two Normal distributions when variances are unknown

22.2 Assumptions

  • \(\large (x_{11}, x_{12},...,x_{1n_1})\) is a random sample of size \(\large n_1\) from Population 1

  • \(\large (x_{21}, x_{22},...,x_{2n_2})\) is a random sample of size \(\large n_2\) from Population 2

  • The two populations are independent

  • The variable is distributed as a Normal distribution for both populations with unknown variances


22.3 Scenarios

  • Scenario 1: Two independent normal populations with unknown means \(\large \mu_1\) and \(\large \mu_2\), and unknown but equal variances, \(\large \sigma_1^2 = \sigma_2^2 = \sigma^2\)

  • Scenario 1: Two independent normal populations with unknown means \(\large \mu_1\) and \(\large \mu_2\), and unknown but uequal variances, \(\large \sigma_1^2 \ne \sigma_2^2\)


22.4 Steps of Hypothesis testing

  1. Identify the parameter of interest

  2. Define \(\large H_O\) and \(\large H_A\)

  3. Define a significance level \(\large \alpha\)

  4. Calculate an estimate of the parameter

  5. Determine an appropriate test statistic, its distribution when \(\large H_O\) is correct, calculate the value of test statistic from the sample

  6. Obtain the probability under the distribution of the test statistic

  7. Compare the observed probability given \(\large \alpha\) and conclude