Let's work through the problem based on the image you uploaded.

Step 1: Calculate the correlation coefficient (r)

The correlation coefficient $r$ can be calculated using the formula:

[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} ]

Where:

• $n$ is the number of data points
• $x$ and $y$ are the respective data points for the variables

For the given dataLet's work through the problem based on the image you uploaded.

Step 1: Calculate the correlation coefficient (r)

The correlation coefficient $r$ can be calculated using the formula:

[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} ]

Where:

• $n$ is the number of data points
• $x$ and $y$ are the respective data points for the variables

For the 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 will calculate these values for you.

Step 2: Hypothesis Testing

Next, we'll perform a hypothesis test at a significance level of $\alpha = 0.01$.

Null hypothesis ($H_0$): $\rho = 0$

There is no significant relationship between the year and the high temperature.

Alternative hypothesis ($H_a$): $\rho \neq 0$

There is a significant relationship between the year and the high temperature.

For this, we'll:

• Calculate the test statistic $t$
• Determine the p-value based on $t$

Let me first calculate the correlation coefficient $r$. I'll work on that.The calculated correlation coefficient $r = 0.970$ (rounded to three decimal places).

Step 2: Testing the Significance of the Correlation

Now, we will test the significance of this correlation coefficient using the given significance level $\alpha = 0.01$.

The test statistic $t$ for testing the correlation is given by:

$t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}}$

Let me calculate the test statistic $t$.The calculated test statistic is $t = 9.778$ (rounded to three decimal places).

Step 3: Calculate the p-value

To complete the hypothesis test, we need to calculate the