Section 23 Hypothesis Testing - Two Samples: Steps

23.1 Two Samples, Unknown Variance

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

23.2 Steps

1. Identify the parameter of interest: Population mean \(\large \mu\)

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

\[\large H_O: \mu_1 = \mu_2\]

\[\large H_A: \mu_1 \ne \mu_2\]

3. Define \(\large \alpha\)

\[\large \alpha = 0.05\]

4. Calculate an estimate of the parameter

Sample Mean: \[ \large \bar{x_1} = \frac{1}{n}\sum\limits_{i=1}^{n} x_{1i} \] \[ \large \bar{x_2} = \frac{1}{n}\sum\limits_{i=1}^{n} x_{2i} \]

Pooled Sample Standard Deviation:

Scenario 1: Variances of TWO samples are equal

\[ \large s = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}\]

Scenario 2: Variances of TWO samples are NOT equal

\[ \large s = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]

Where: \(\large s_1^2\) and \(\large s_2^2\) are variances from Sample 1 and Sample 2, respectively.

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

\[ \large t_{Cal} = \frac{(\bar{x_1}-\bar{x_2}) - (\mu_1 - \mu_2)} {s\sqrt{1/n_1+1/n_2}} \]

\[ \large t_{Cal} = \frac{(\bar{x_1}-\bar{x_2})} {s\sqrt{1/n_1+1/n_2}} \]

• Note - The test statistic is the difference of Observed Difference & Expected Difference - The test statistic represents the ratio of signal to error - The test statistic is centred and scaled

Distribution of the test statistic

\[\large t \hspace{6mm} distribution \hspace{6mm} with \hspace{6mm} (n_1+n_2-2) \hspace{2mm} df \]

6. Obtain the probability under the distribution of the test statistic (two-tailed probability)

\[\large 2*pt(q = |t_{Cal}|, df = n1+n2-2, lower.tail=FALSE)\]

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

Find the probability that \(\large t\) is less than or equal to \(\large |t_{Cal}|\) from \(\large t\) distribution with \(\large (n_1+n_2-2)\) degrees of freedom.