In Exercises \(14.128-14.133\). we repeat the information from Exercises \(14.22-14.27\) Presuming that the assumptions for regression inferences are met, perform the required correlation \(t-\)tests, using either the critical- value approach or the \(P-\)value approach.

Following are the data size and price data for custom homes from Exercise \(14.24\).

At the \(0.5%\) significance level, do the data provide sufficient evidence to conclude that, for custom homes in the Equestrian Estates, size and price are positively linearly correlated?

The level of significance is \(0.005\) and the data is,

The hypothesis are,

\(H_{0}:\rho=0\)

\(H_{a}:\rho<0\)

The table is shown below.

The value of \(r\) is,

\(r=\frac{\sum x_{i}y_{i}-\sum x_{i}\sum \frac{y_{i}}{n}}{\sqrt{\sum x_{i}^{2}-(\sum x_{i}^{2})}\sqrt{\sum y_{i}^{2}-(\sum y_{i})^{2}}}\)

\(=\frac{169993-(270)\left ( \frac{5552}{9} \right )}{\sqrt{8316-\frac{(270)^{2}}{9}}\sqrt{3504412-\frac{(5552)^{2}}{9}}}\)

\(=0.828731691\)

The value of test statistic is,

\(t=\frac{r}{\sqrt{\frac{1-r^{2}}{n-2}}}\)

\(=\frac{0.828731691}{\sqrt{\frac{1-(0.828731691)^{2}}{9-2}}}\)

\(=3.92\)

The degree of freedom is,

\(dof=n-2\)

\(=9-2\)

\(=7\)

The curve is shown below.

Since, the value lies in the rejection region.

Thus, the null hypothesis is rejected and it can be conclude that for custom homes in the Equestrian Estate, size and price are positively linearly correlated.