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 on percentage of investments in energy securities and tax efficiency from Exercise \(14.22\). At the \(2.5%\) significance level, do the data provide sufficient evidence to conclude that percentage of investments in energy securities and tax efficiency are negatively linearly correlated for mutual fund portfolios?

The level of significance is \(0.025\) 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,

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

\(=\frac{4376.95-(55.9)\left ( \frac{832.5}{10} \right )}{\sqrt{365-\frac{(55.9)^{2}}{10}}\sqrt{70838.49-\frac{(832.5)^{2}}{10}}}\)

\(=-0.974836057\)

The value of test statistic is,

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

\(=\frac{-0.974836057}{\sqrt{\frac{1-(-0.974836057)^{2}}{3-2}}}\)

\(=-0.12.42\)

The degree of freedom is,

\(dof=n-2\)

\(=10-2\)

\(=8\)

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 percentage of investment in energy securities and tax efficient are negatively linearly correlated for mutual fund portfolios