A Breeder of Thoroughbred Horses Wishes to Model the Relationship

A breeder of Thoroughbred horses wishes to model the relationship between the gestation period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. $\begin{array} { c c c c c c } \text { Horse } & \begin{array} { c } \text { Gestation } \\ \text { period } \end{array} & \begin{array} { c } \text { Life } \\ \text { Length } \end{array} & \text { Horse } & \begin{array} { c } \text { Gestation } \\ \text { period } \end{array} & \begin{array} { c } \text { Life } \\ \text { Length } \end{array} \\ & x \text { (days) } & y \text { (years) } & & x \text { (days) } & y \text { (years) } \\ 1 & 416 & 24 & 5 & 356 & 22 \\ 2 & 279 & 25.5 & 6 & 403 & 23.5 \\ 3 & 298 & 20 & 7 & 265 & 21 \\ 4 & 307 & 21.5 & & & \end{array}$

Summary statistics yield $S S _ { x x } = 21,752 , S S _ { x y } = 236.5 , S S _ { y y } = 22 , \bar { x } = 332$ , and $\bar { y } = 22.5$ . Test to determine if a linear relationship exists between the gestation period and the length of life of a horse. Use $\alpha = .05$ and use $s = 1.97$ as an estimate of $\sigma$ .