Let's walk through this step by step:

1. Calculate the correlation coefficient $r$: The formula for the Pearson correlation coefficient $r$ is: [ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} ] We will need to calculate the necessary sums:

   • $\sum x$, $\sum y$
   • $\sum x^2$, $\sum y^2$
   • $\sum xy$

2. Test statistic: To test the significance of the correlation coefficient, we use the formula for the test statistic $t$: $t = \frac{r \sqrt{n-2}}{\sqrt{1 - r^2}}$ where $n$ is the number of data points, and $r$ is the correlation coefficient calculated earlier.

3. P-value: Using the test statistic $t$, we can calculate the p-value by referring to the t-distribution table with $n-2$ degrees of freedom.

4. Decision: Based on the p-value, if it is less than the significance level $\alpha = 0.01$, we reject the null hypothesis $H_0$, otherwise we fail to reject it.

Now, let's calculate these values step by step.

Data given:

• $x = \{3, 4, 5, 6, 7, 8, 9, 10\}$
• $y = \{7.4, 11.5, 13.2, 16.6, 17.2, 18.4, 20.2, 20.9\}$

I'll start calculating the necessary sums.The calculated correlation coefficient $r$ is 0.970 (rounded to three decimal places).

Now, let's move on to calculating the test statistic using the formula: $t = \frac{r \sqrt{n-2}}{\sqrt{1 - r^2}}$ where $n = 8$ and $r = 0.970$.