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  • STA258 Lecture 10 Raw

Paired Samples

• Example:
  • We have test 1 and two sets of students.
  • Now we include test 2:
  • $\begin{array}{c}\text{Paired Data}\\ \text{Test 1}& \text{Test 2}\\ X_1& X_1^{\prime }\\ X_2& X_2^{\prime }\\ \dots & \dots \\ X_n& X_n^{\prime }\end{array}$
  • The students are independent, but the same student, $X_1$ influences $X_1^{\prime }$.
  • We can't compare ${\mu }_{t_2}$ with ${\mu }_{t_1}$
  • We can consider the difference between $X_1^{\prime },X_1$ instead.
  • $\begin{array}{c}\text{Paired Data}\\ \text{Test 1}& \text{Test 2}\\ \underset{―}{X_1}& \underset{―}{X_1^{\prime }}& X_1-X_1^{\prime }\\ X_2& X_2^{\prime }\\ \dots & \dots \\ X_n& X_n^{\prime }\end{array}$
  • Now it makes it independent. $X_n-X_n^{\prime }$ vs ${\overline{X}}_{t_1}-{\overline{X}}_{t_2}$
  • $CI={\overline{X}}_d\pm t_{(\nu )}^{\star }\cdot \mathrm{SE}[{\overline{X}}_d]$
  • $CI={\overline{X}}_d\pm t_{(\nu )}^{\star }\cdot \frac{S_d}{\sqrt{n}}$
  • $\nu =n-1$ because we have $n$ independent pairs.