1. Paired two sample procedures

[describe how this situation simplifies to the one sample t-test scenario] Paired data, transformed to new variable representing the difference, sample size is the number of pairs/differences

\[SE(\bar{d}) = \sqrt{\frac{s_d^2}{n}}\]

Assumptions and conditions

  • Paired assumption

    • The two samples are strongly dependent in a paired manner. Specifically, each data point from one sample has a corresponding paired data point in the other sample. This typically is the case when our data represent before and after measurements on the same observational unit. Other examples include:

    • the IQ scores of twins,

    • reaction times with dominant and non-dominant hands,

    • happiness levels of married couples.

  • Independence assumption

    • Each observed difference (between a pair of data points) is independent of the other differences

  • Randomization condition

  • Nearly Normal Condition or the number of pairs is large enough for the CLT to approximate the sampling distribution of \(\bar{x}_1\) and \(\bar{x}_2\).

Paired t-interval

\[\bar{d} \pm \left[t^*_{a, n-1} \times SE(\bar{d}) \right],\] where \(t^*_a\) is the \(\left(\frac{1-a}{2}\right)^{th}\) lower quantile of a Student’s t-distribution with \((n-1)\) degrees of freedom.

Paired t-test

Let \(\mu_d = \mu_1 - \mu_2\) be the unknown true difference in the means of two dependent populations.

\[H_0: \mu_d = \Delta_0\]

\[T.S. = \frac{\bar{d} - \Delta_0}{SE(\bar{d})} \stackrel{H_0}{\sim} \text{Student's t dist with $(n-1)$ degrees of freedom}\] # 2. Looking ahead

Instructions for creating your final project poster have been shared on Moodle.