The Data for a Random Sample of Six Paired Observations

The data for a random sample of six paired observations are shown below. $\begin{array} { c c c } \text { Pair } & \text { Observation 1 } & \text { Observation 2 } \\ 1 & 1 & 3 \\ 2 & 2 & 4 \\ 3 & 3 & 5 \\ 4 & 4 & 6 \\ 5 & 5 & 7 \\ 6 & 6 & 8 \end{array}$
a. Calculate the difference between each pair of observations by subtracting observation 2 from observation 1. Use the differences to calculate $s d ^ { 2 }$ .
b. Calculate the standard deviations $s _ { 1 } ^ { 2 }$ and $s _ { 2 } ^ { 2 }$ of each column of observations. Then find pooled estimate of the variance $s p ^ { 2 }$ .
c. Comparing $s ^ { 2 } d ^ { 2 }$ and $s p ^ { 2 }$ , explain the benefit of a paired difference experiment.