Consider $n=5$ pairs $(x_1,y_1),\ldots,(x_n,y_n)$. Let ${\overline x}=n^{-1}\sum_{j=1}^n x_j$ and ${\overline y}=n^{-1}\sum_{j=1}^n y_j$ be the sample means of the $x$ and $y$ variables. Let $s_x$ and $s_y$ be the corresponding standard deviations. Let $s_{xy}$ and $r_{xy}$ be the sample covariance and sample correlation respectively. Suppose ${\overline x}=5.4$, ${\overline y}=7.2$, $s_x=2.702$, $s_y=3.701$, $s_{xy}=9.65$.
Part a)
What is the sample correlation of the $(x_i,y_i)$?

For parts (b) to (e), consider a linearly transformed variable $x^*_i=3+3\cdot x_i$ for $i=1,\ldots,n$.
Part b)
What is ${\overline x}^*$?

Part c)
What is $s_{x^*}$?

Part d)
What is the sample covariance of the $(x_i^*,y_i)$?

Part e)
What is the sample correlation of the $(x_i^*,y_i)$?

Part f)
Consider a quadratic transformed variable $x^{**}_i=3+3\cdot x_i+x_i^2$ for $i=1,\ldots,n$. As above $n=5$, ${\overline x}=5.4$ and $s_x=2.702$.
What is ${\overline x}^{**}$, the sample mean of the transformed variable?

Hint: