To find the equation of the regression line $\hat{y} = b_0 + b_1x$ for predicting the number of bids ($\hat{y}$) based on the list price ($x$), we need to calculate the slope ($b_1$) and the y-intercept ($b_0$) of the regression line. Here are the steps involved:

Step 1: Calculate the Means

Calculate the mean of the list prices ($\overline{x}$) and the mean of the number of bids ($\overline{y}$).

$\overline{x} = \frac{23 + 24 + 32 + 37 + 49}{5}$

$\overline{y} = \frac{1 + 2 + 6 + 9 + 10}{5}$

Step 2: Calculate the Slope ($b_1$)

The slope $b_1$ is calculated using the formula:

$b_1 = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2}$

Step 3: Calculate the Y-intercept ($b_0$)

The y-intercept $b_0$ is calculated using the formula:

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

Step 4: Compute the Correlation Coefficient (Optional)

To determine if the regression line is statistically significant, you may calculate the correlation coefficient $r$ using:

[ r = \frac{n\sum(x_i y_i) - \sum x_i \sum y_i}{\sqrt{[n\sum x_i^2 - (\sum x_i)^2][n\sum y_i^2 - (\sum y_i)^2]}} ]

Let's go through these calculations step by step.

Step 1: Calculate the Means

$\overline{x} = \frac{23 + 24 + 32 + 37 + 49}{5} = \frac{165}{5} = 33$

$\overline{y} = \frac{1 + 2 + 6 + 9 + 10}{5} = \frac{28}{5} = 5.6$

Step 2: Calculate the Slope ($b_1$)

First, calculate $(x_i - \overline{x})$ and $(y_i - \overline{y})$ for each data point:

[ (x_i - \overline{x}) = [23-33, 24-33, 32-33, 37-33, 49-33] = [-10, -9, -1, 4, 16] ]

[ (y_i - \overline{y}) = [1-5.6, 2-5.6, 6-5.6, 9-5.6, 10-5.6] = [-4.6, -3.6, 0.4, 3.4, 4.4] ]

Next, calculate the products $(x_i - \overline{x})(y_i - \overline{y})$:

[ [-10 \times -4.6, -9 \times -3.6, -1 \times 0.4, 4 \times 3.4, 16 \times 4.4] = [46, 32.4, -0.4, 13.6, 70.4] ]

Sum of the products:

$\sum (x_i - \overline{x})(y_i - \overline{y}) = 46 + 32.4 - 0.4 + 13.6 + 70.4 = 162$

Now, calculate $\sum (x_i - \overline{x})^2$:

$(-10)^2 + (-9)^2 + (-1)^2 + 4^2 + 16^2 = 100 + 81 + 1 + 16 + 256 = 454$

Finally, calculate $b_1$:

$b_1 = \frac{162}{454} \approx 0.357$

Step 3: Calculate the Y-intercept ($b_0$)

$b_0 = \overline{y} - b_1\overline{x} = 5.6 - 0.357 \times 33 \approx 5.6 - 11.781 \approx -6.181$

Step 4: Equation of the Regression Line

$\hat{y} = -6.181 + 0.357x$

Step 5: Correlation Coefficient (Optional)

This requires further calculation to determine if the correlation coefficient is statistically significant, which would indicate whether it's appropriate to use this regression line for prediction.

Would you like to continue with calculating the correlation coefficient, or do you have other questions related to this regression analysis?

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Related Questions:

1. How is the correlation coefficient calculated in this context?
2. What does a positive or negative slope in a regression line indicate?
3. How do you interpret the y-intercept in this regression model?
4. What are the implications if the correlation coefficient is not statistically significant?
5. How can you test the significance of the correlation coefficient?
6. What is the meaning of the term "least squares" in regression analysis?
7. How does the regression line minimize the sum of squared errors?
8. Can you apply the regression model to data outside the given range of prices?

Tip: Always check the statistical significance of the correlation coefficient before using the regression line to make predictions.