Question

For each exercise, decide, at 10% significance level, whether the data provide sufficient evidence t reject the null hypothesis in favour of alternate hypothesis

Short answer

We accept the null hypothesis that variables are not linearly (positively) correlated.

Step-by-step solution

r calculation 

Sample correlation coefficientr is calculated : $r=\sum _{}zxzy/(n-1)$

where $z{x}_n=(x_n-xmean)\hspace{0.17em}/s\tan darddeviation$, and same for $z{y}_n$

Using excel and above formula concept , r = 0.5

Hypothesis Testing 

r positive ( $r>0$) for sample needs to be tested using hypothesis testing, to ensure that the population correlation parameter p(rho) is positive

$$\begin{gathered} t=r/\sqrt{(1-r^2)\hspace{0.17em}/(n-2)\hspace{0.17em}} \\ =0.5/\sqrt{(1-0.25)/(3-2)\hspace{0.17em}} \\ =0.5/\sqrt{0.75} \\ =0.5/0.8660 \end{gathered}$$

$=0.577$

As calculated t value tabulated t at 0.10 significance level, so p value would be greater at the same significance level.

So, accept the null hypothesis and conclude that variables are not linearly correlated.