The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x
, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Price in Dollars	29
33
34
36
46
Number of Bids	1
3
8
9
10
Table

Step 2 of 6 :  Find the estimated y-intercept. Round your answer to three decimal places.

To find the estimated y-intercept \( b_0 \) for the regression line \( \hat{y} = b_0 + b_1x \), we first need to compute the slope \( b_1 \) and use the formula for the y-intercept:

\[
b_0 = \bar{y} - b_1 \bar{x}
\]

Where:
- \( \bar{y} \) is the mean of the number of bids.
- \( \bar{x} \) is the mean of the list prices.
- \( b_1 \) is the slope of the regression line, which is calculated as:

\[
b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]

### Step 1: Compute the means
We have the following data:

**List prices** (\( x \)): 29, 33, 34, 36, 46  
**Number of bids** (\( y \)): 1, 3, 8, 9, 10

First, let's calculate the means of \( x \) and \( y \).

\[
\bar{x} = \frac{29 + 33 + 34 + 36 + 46}{5} = \frac{178}{5} = 35.6
\]
\[
\bar{y} = \frac{1 + 3 + 8 + 9 + 10}{5} = \frac{31}{5} = 6.2
\]

### Step 2: Calculate \( b_1 \)
Now, we calculate the numerator and denominator for the slope \( b_1 \).

1. Compute the deviations \( (x_i - \bar{x}) \) and \( (y_i - \bar{y}) \), and their products:

| \( x_i \) | \( y_i \) | \( x_i - \bar{x} \) | \( y_i - \bar{y} \) | \( (x_i - \bar{x})(y_i - \bar{y}) \) | \( (x_i - \bar{x})^2 \) |
|----------|----------|---------------------|---------------------|-------------------------------------|-----------------------|
| 29       | 1        | -6.6                | -5.2                | 34.32                               | 43.56                 |
| 33       | 3        | -2.6                | -3.2                | 8.32                                | 6.76                  |
| 34       | 8        | -1.6                | 1.8                 | -2.88                               | 2.56                  |
| 36       | 9        | 0.4                 | 2.8                 | 1.12                                | 0.16                  |
| 46       | 10       | 10.4                | 3.8                 | 39.52                               | 108.16                |

Now sum the columns for \( (x_i - \bar{x})(y_i - \bar{y}) \) and \( (x_i - \bar{x})^2 \):

\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 34.32 + 8.32 - 2.88 + 1.12 + 39.52 = 80.4
\]

\[
\sum (x_i - \bar{x})^2 = 43.56 + 6.76 + 2.56 + 0.16 + 108.16 = 161.2
\]

Now, calculate the slope \( b_1 \):

\[
b_1 = \frac{80.4}{161.2} \approx 0.499
\]

### Step 3: Calculate the y-intercept \( b_0 \)
Finally, use the formula for \( b_0 \):

\[
b_0 = \bar{y} - b_1 \bar{x} = 6.2 - (0.499)(35.6) = 6.2 - 17.74 = -11.54
\]

### Final Answer
The estimated y-intercept \( b_0 \) is approximately **-11.540**.

Would you like to proceed with more details, or do you have any further questions on this? Here are some related questions to deepen the understanding:

1. How would you compute the slope and intercept for a larger dataset?
2. What is the interpretation of the regression equation in this context?
3. How do you test the significance of the correlation coefficient?
4. What would happen if the correlation coefficient were zero?
5. How can residuals help assess the quality of the regression model?

**Tip**: Always check the assumptions of linear regression (linearity, independence, homoscedasticity, and normality of residuals) before making predictions.

The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Price in Dollars: 29, 33, 34, 36, 46
Number of Bids: 1, 3, 8, 9, 10
Step 2 of 6: Find the estimated y-intercept. Round your answer to three decimal places.

Learn how to calculate the estimated y-intercept for the regression line predicting the number of bids based on list price using a set of auction data. This step-by-step solution involves linear regression and the Least Squares method, suitable for students in Grades 10-12.

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