Question

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

Ha :$pnot=0$.

x	0	4	3	1	2
y	1	9	8	4	3

Short answer

We don't reject Ha : $pnot=0$, and state that the variables are linearly correlated.

Step-by-step solution

Correlation Coefficient 

Sample correlation coefficient is calculated by : $r=\sum _{}^{}zxzy/(n-1)\hspace{0.17em}$

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

Using excel and above formula concept,$r=0.9325$

Hypothesis Testing 

$rnot=0$for sample needs to be tested using hypothesis testing, to ensure that the population correlation parameterp (rho) is non zero.

$$\begin{gathered} t=r/\sqrt{(1-r^2)\hspace{0.17em}/(n-2)} \\ 0.9325/\sqrt{(1-0.9325)/(5-2)} \\ 0.9325/\sqrt{0.0675/3} \\ 0.9325/\sqrt{0.0225} \\ 0.9325/0.15 \\ t=6.216 \end{gathered}$$

As t calculated value is greater than t value for $10\%$significance level ie $0.05$each side, and p value $=0.02$is also lesser than p at the same significance level

So, we reject the null hypothesis H0 : $p=0$in favour of alternate hypothesis H1 : $pnot=0$. Hence, the variables are linearly correlated.