Question

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

$H_a:p\ne 0$

Short answer

The null hypothesis is not rejected and state that the variables are linearly correlated.

Step-by-step solution

Step 1. Given information

The level of significance is $0.1$and the data is,

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

Step 2. Calculation

The hypothesis are,

$$\begin{gathered} H_0:\rho =0 \\ {H}_a:\rho \ne 0 \end{gathered}$$

The table is shown below.

The value of $r$is,

$$\begin{gathered} r=\frac{\sum _{}{x}_i{y}_i-\sum _{}{x}_i\sum _{}{\displaystyle \frac{y_i}{n}}}{\sqrt{\sum _{}{x}_i^2-{\left(\sum _{}{x}_i\right)}^2}\sqrt{\sum _{}{y}_i^2-{\left(\sum _{}{y}_i\right)}^2}} \\ =\frac{70-\left(10\right)\left({\displaystyle \frac{25}{5}}\right)}{\sqrt{30-{\displaystyle \frac{{\left(10\right)}^2}{5}}}\sqrt{171-{\displaystyle \frac{{\left(25\right)}^2}{5}}}} \\ =0.933 \end{gathered}$$

The value of test statistic is,

$$\begin{gathered} t=\frac{r}{\sqrt{{\displaystyle \frac{1-r^2}{n-2}}}} \\ =\frac{0.933}{\sqrt{{\displaystyle \frac{1-{(0.933)}^2}{5-2}}}} \\ =4.49 \end{gathered}$$

The degree of freedom is,

$$\begin{gathered} dof=n-2 \\ =5-2 \\ =3 \end{gathered}$$

The curve is shown below.

Since, the value do not lie in the rejection region.

Thus, the null hypothesis is not rejected.