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	Number of Bids
28	2
29	3
35	4
36	6
50	7 Find the value of the coefficient of determination. Round your answer to three decimal places.

To find the coefficient of determination (\( R^2 \)), we first need to calculate the correlation coefficient (\( r \)) and then square it. Here's the step-by-step process:

### 1. Data:
| Price (\( x \)) | Bids (\( y \)) |
|:---------------:|:--------------:|
| 28              | 2              |
| 29              | 3              |
| 35              | 4              |
| 36              | 6              |
| 50              | 7              |

### 2. Calculating the correlation coefficient (\( r \)):
The formula for the correlation coefficient is:

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

Where:
- \( n \) is the number of data points (5 in this case),
- \( x \) represents the list prices, and
- \( y \) represents the number of bids.

### 3. Summations:
We need to calculate the following sums for the formula:
- \( \sum{x} \)
- \( \sum{y} \)
- \( \sum{xy} \)
- \( \sum{x^2} \)
- \( \sum{y^2} \)

I'll now compute the necessary values and the correlation coefficient.The correlation coefficient \( r \) is approximately \( 0.902 \).

The coefficient of determination \( R^2 \), which is the square of \( r \), is approximately \( 0.814 \) (rounded to three decimal places).

This means that about 81.4% of the variation in the number of bids can be explained by the list price of the items.

Would you like more details on how this result was obtained or how to interpret it?

Here are five follow-up questions:
1. What does a high \( R^2 \) value indicate about the strength of a regression model?
2. How would you test the statistical significance of the correlation coefficient?
3. Could there be outliers affecting this result? How would you check for them?
4. What are the assumptions behind using linear regression for prediction?
5. How would you compute the regression line equation from the data?

**Tip:** A high \( R^2 \) value suggests a strong relationship between variables, but it's important to check residuals to ensure the model is appropriate.

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: 28, 29, 35, 36, 50. Number of Bids: 2, 3, 4, 6, 7. Find the value of the coefficient of determination. Round your answer to three decimal places.

Learn how to calculate the coefficient of determination (R²) for predicting the number of bids on items sold through online auctions based on list price. This example includes list prices and bids data and involves regression analysis, the correlation coefficient, and R². Suitable for Grades 10-12.

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