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+b1xy^=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

2929

3333

3434

3636

4646

Number of Bids

11

33

88

99

1010

Table

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Step 6 of 6 :

Find the value of the coefficient of determination. Round your answer to three decimal places.

To find the coefficient of determination (denoted as \( R^2 \)) for the given data, we first need to compute the correlation coefficient \( r \) and then square it to obtain \( R^2 \). Here’s how we can do that:

### Step 1: Organize the Data
We have the following data points:

| Price (X) | Number of Bids (Y) |
|-----------|---------------------|
| 2929      | 11                  |
| 3333      | 33                  |
| 3434      | 88                  |
| 3636      | 99                  |
| 4646      | 1010                |

### Step 2: Calculate the Correlation Coefficient \( r \)

The correlation coefficient \( r \) can be calculated using the formula:

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

Where:
- \( n \) = number of data points
- \( \sum xy \) = sum of the products of each pair
- \( \sum x \) = sum of all \( x \) values
- \( \sum y \) = sum of all \( y \) values
- \( \sum x^2 \) = sum of squares of \( x \) values
- \( \sum y^2 \) = sum of squares of \( y \) values

### Step 3: Calculate Each Component

1. **Calculate sums and products:**

   \[
   \sum x = 2929 + 3333 + 3434 + 3636 + 4646 = 17978
   \]
   \[
   \sum y = 11 + 33 + 88 + 99 + 1010 = 1241
   \]
   \[
   \sum xy = (2929 \times 11) + (3333 \times 33) + (3434 \times 88) + (3636 \times 99) + (4646 \times 1010)
   \]
   \[
   \sum x^2 = 2929^2 + 3333^2 + 3434^2 + 3636^2 + 4646^2
   \]
   \[
   \sum y^2 = 11^2 + 33^2 + 88^2 + 99^2 + 1010^2
   \]

2. **Calculate \( n \)**:
   \[
   n = 5
   \]

### Step 4: Plug the Values into the Formula

Now, let's calculate \( r \) and then \( R^2 \). I'll perform these calculations.The coefficient of determination \( R^2 \) for the given data is approximately **0.891** (rounded to three decimal places). 

This value indicates that about 89.1% of the variability in the number of bids can be explained by the list price of the items.

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. What does the coefficient of determination imply about the relationship between price and number of bids?
2. How would you interpret a low \( R^2 \) value in this context?
3. What are the assumptions of linear regression analysis?
4. How can you check if the correlation coefficient is statistically significant?
5. What other factors might influence the number of bids in online auctions? 

**Tip:** Always visualize your data with a scatter plot to better understand the relationship between variables before performing regression analysis.

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. Find the value of the coefficient of determination. Round your answer to three decimal places.

Calculate the coefficient of determination (R^2) for a regression line based on online auction data. Learn how list price affects the number of bids, using linear regression and correlation. This step-by-step guide includes key statistics concepts and is suitable for undergraduate students.

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