Consider a study in which we wish to see whether the lengths of fish are related to the size of the lake in which they live. Unfortunately, the amount of dissolved nitrogen in the water from farm runoff is also thought to influence fish size. Imagine we are provided with the data shown in the table and wish to perform a matched design study where we compare the sizes of the fish in the largest four lakes to those in the smallest four lakes.

\begin{array}{cccc}\text { Fish }& \text { Fish }&\text { Lake }&\text { Dissolved }\\\underline { \text { ID \# } } &\underline { \text { size(cm) } } &\underline { \text { size } } &\underline { \text { Nitrogen } } \\1 & 25 & \text { small } & 3 \\2 & 45 & \text { small } & 21 \\3 & 28 & \text { large } & 4 \\4 & 60 & \text { large } & 15 \\5 & 35 & \text { small } & 8 \\6 & 50 & \text { large } & 20 \\7 & 33 & \text { small } & 14 \\8 & 40 & \text { large } & 7\end{array}

(a) Which fish should be in each of the two observation groups, and how should they be matched?
(b) After matching, what is the mean and standard error of the differences in the sizes of the fish?
(c) Based on your values in part (b), does there seem to be a difference in the mean size of the fish in the small and large lakes?

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