The following table gives the data for the average temperature and the snow accumulation in several small towns for a single month. Determine the equation of the regression line, yˆ=b0+b1x. Round the slope and y-intercept to the nearest thousandth. Then determine if the regression equation is appropriate for making predictions at the 0.01 level of significance.

Critical Values of the Pearson Correlation Coefficient

Average Temperatures and Snow AccumulationsAverage Temperature (℉)

39

34

19

41

44

22

25

20

25

42

Snow Accumulation (in.)

10

16

28

10

14

25

26

16

14

10

To solve this problem, we need to:

1. **Find the equation of the regression line** \( \hat{y} = b_0 + b_1x \), where:
   - \( b_1 \) is the slope of the regression line.
   - \( b_0 \) is the y-intercept.

2. **Determine if the regression equation is appropriate for making predictions at the 0.01 level of significance**, using the Pearson correlation coefficient \( r \) and comparing it to critical values.

### Step 1: Data summary

Here is the dataTo solve this problem, we need to:

1. **Find the equation of the regression line** \( \hat{y} = b_0 + b_1x \), where:
   - \( b_1 \) is the slope of the regression line.
   - \( b_0 \) is the y-intercept.

2. **Determine if the regression equation is appropriate for making predictions at the 0.01 level of significance**, using the Pearson correlation coefficient \( r \) and comparing it to critical values.

### Step 1: Data summary

Here is the 

- **Average Temperature (x)**: \( [39, 34, 19, 41, 44, 22, 25, 20, 25, 42] \)
- **Snow Accumulation (y)**: \( [10, 16, 28, 10, 14, 25, 26, 16, 14, 10] \)

We will now calculate the slope \( b_1 \), the intercept \( b_0 \), and the correlation coefficient \( r \) using statistical formulas for linear regression.

### Step 2: Calculating the slope \( b_1 \) and intercept \( b_0 \)

The formulas for linear regression coefficients are:

\[
b_1 = \frac{n\sum(xy) - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
\[
b_0 = \frac{\sum y - b_1 \sum x}{n}
\]

Where:
- \( x \) = average temperature
- \( y \) = snow accumulation
- \( n \) = number of data points (here \( n = 10 \))

I'll calculate this now.The equation of the regression line is:

\[
\hat{y} = 33.631 - 0.538x
\]

### Step 3: Determine if the regression equation is appropriate for predictions at the 0.01 level of significance

To do this, we need to compute the **Pearson correlation coefficient** \( r \) and compare it with the critical values for the significance level \( \alpha = 0.01 \). 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 now compute \( r \) and check its significance.The Pearson correlation coefficient \( r = -0.767 \).

### Step 4: Checking significance at the 0.01 level

To determine if the regression equation is appropriate for making predictions, we compare the absolute value of \( r \) with the critical value for the Pearson correlation coefficient at \( n - 2 = 8 \) degrees of freedom for \( \alpha = 0.01 \). The critical value for \( \alpha = 0.01 \) with 8 degrees of freedom is approximately **0.765**.

Since \( |r| = 0.767 \) is slightly greater than the critical value of 0.765, the correlation is statistically significant at the 0.01 level. Therefore, the regression equation is appropriate for making predictions at this level of significance.

---

Would you like to go over any specific details or calculations? Here are five related questions:

1. How do you interpret the slope in the context of temperature and snow accumulation?
2. What is the meaning of the correlation coefficient being negative?
3. How would predictions differ if the correlation coefficient were weaker?
4. Can the regression line be used for all temperature values, or are there limitations?
5. How does the significance level affect the reliability of the regression model?

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

The following table gives the data for the average temperature and the snow accumulation in several small towns for a single month. Determine the equation of the regression line, yˆ=b0+b1x. Round the slope and y-intercept to the nearest thousandth. Then determine if the regression equation is appropriate for making predictions at the 0.01 level of significance.

Average Temperatures and Snow Accumulations

Average Temperature (℉): [39, 34, 19, 41, 44, 22, 25, 20, 25, 42]
Snow Accumulation (in.): [10, 16, 28, 10, 14, 25, 26, 16, 14, 10]

Learn how to find the regression line for temperature and snow accumulation data, including calculating the slope, y-intercept, and Pearson correlation coefficient. This guide explains whether the regression equation is statistically significant at the 0.01 level, using real data from small towns.

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