To determine the equation of the regression line, we first need to calculate the slope ($b_1$) and the intercept ($b_0$) of the line, as well as test if the regression equation is statistically significant at the $0.05$ level of significance.

Step 1: Organize Data

Let $x$ be the Average Temperature (℉) and $y$ be the Snow Accumulation (in.).

• $x = [39, 29, 20, 44, 45, 15, 34, 25, 28, 42]$
• $y = [7, 19, 22, 10, 14, 29, 22, 19, 20, 5]$

Step 2: Calculate the Slope and Intercept

The formula for the slope ($b_1$) is:

$b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

The formula for the intercept ($b_0$) is:

$b_0 = \bar{y} - b_1 \bar{x}$

Where:

• $n$ is the number of data points (here, $n = 10$),
• $\sum xy$ is the sum of the products of $x$ and $y$,
• $\sum x$ and $\sum y$ are the sums of $x$ and $y$ respectively,
• $\sum x^2$ is the sum of the squares of $x$,
• $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$.

Let's calculate these values.The equation of the regression line is:

$\hat{y} = 36.454 - 0.615x$

Now, we need to determine if the regression equation is appropriate for making predictions at the $0.05$ level of significance by testing the significance of the slope using the Pearson correlation coefficient ($r$).

Step 3: Calculate the Pearson Correlation Coefficient ($r$)

The formula for $r$ 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 also need to check the critical values of $r$ from a table (depending on $n-2 = 8$ degrees of freedom). Let's calculate the correlation coefficient $r$.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like help with the Pearson correlation coefficient calculation manually, or any further clarification about the regression process?

Here are 5 follow-up questions you might find interesting:

1. How is the Pearson correlation coefficient interpreted?
2. What does it mean if the slope of the regression line is close to zero?
3. How can we test the hypothesis for the significance of the slope in a regression?
4. What is the difference between simple linear regression and multiple linear regression?
5. How do we interpret the R-squared value in a regression analysis?

Tip: Always check the assumptions of linear regression before making predictions, such as the linearity of the relationship and homoscedasticity of residuals.