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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.

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### 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 \( r = 0.970 \).

Annual high temperatures in a certain location have been tracked for 8 years. Calculate the correlation coefficient and test its significance at a 0.01 significance level.

This problem involves calculating the Pearson correlation coefficient for 8 years of high temperature data and testing its significance at a 0.01 level. Key concepts include correlation, hypothesis testing, and using t-distribution to determine the p-value.

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