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.

$\begin{array}{lllll}\text { Horse}&\text { Gestation}&\text { Life }&\text { Horse }&\text { Gestation }&\text { Life }\\&\text { period }&\text { Length }&&\text { period } &\text { Length }\\&x \text { (days) } &y \text { (years) } && x \text { (days) } &y \text { (years) }\\1&416 & 24 & 5 & 356 & 22 \\2&279 & 25.5 & 6 & 403 & 23.5 \\3&298 & 20 & 7 & 265 & 21 \\4&307 & 21.5 & & &\end{array}$

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 $\sigma$ .