Inferences for Matched Pairs

Let di = difference between pairs, $\overline{d}$ = average of the pairs, sd = the standard deviation of the differences and nd = the number of differences.

Paired t Test

1.

Null Hypothesis

$H_{0}: \ \mu_{d} = D_{0}$ (D0 specified) This equivalent to $H_{0}: \ \mu_{1} - \mu_{2} = D_{0}$

2.

Alternative Hypothesis

Choose one of the following:

i. $H_{A}: \ \mu_{d} > D_{0}$ or $H_{A}: \ \mu_{1} - \mu_{2} > D_{0}$

ii. $H_{A}: \ \mu_{d} < D_{0}$ or $H_{A}: \ \mu_{1} - \mu_{2} < D_{0}$

iii. $H_{A}: \ \mu_{d} \neq D_{0}$ or $H_{A}: \ \mu_{1} - \mu_{2} \neq D_{0}$

i and ii are one-sided or one-tailed tests. iii is a two-sided or two-tailed test.

3.

T.S. $t = \frac{\overline{d} - D_{0}}{\frac{s_{d}}{\sqrt{n_{d}}}}$

t has nd-1 df.

4.

Calculation of p-value. Depends on on choice of HA.

i. p-value = P(T > t)

ii. p-value = P(T < t)

iii. p-value = 2P(T > |t|)

5.

Draw conclusions. If $\alpha$ decision is to be made, and $\alpha$ is specified, reject H0 if the p-value is less than or equal to $\alpha$.

Confidence Interval for $\mu_{d}$ ( $\mu_{1} - \mu_{2}$)


\begin{displaymath}\overline{d} \pm t^{*}\frac{s_{d}}{\sqrt{n_{d}}} \ \textrm{where $t$ has $n_{d}-1$ df}\end{displaymath}


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Jonathan Payne
1999-03-22