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	21
23
29
43
50
Number of Bids	1
2
4
5
7
Table

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 (\( R^2 \)), we first need to calculate the correlation coefficient (\( r \)) and then square it. The coefficient of determination explains the proportion of the variance in the dependent variable (the number of bids) that can be explained by the independent variable (the list price).

### Step-by-Step Process:

1. **Find the correlation coefficient \( r \)**.
   
   The correlation coefficient can be calculated using the formula:

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

   Where:
   - \( n \) is the number of data points
   - \( x \) is the list price
   - \( y \) is the number of bids
   - \( \sum x \) is the sum of all the \( x \)-values
   - \( \sum y \) is the sum of all the \( y \)-values
   - \( \sum xy \) is the sum of the product of each pair of \( x \) and \( y \)-values
   - \( \sum x^2 \) is the sum of the squares of all the \( x \)-values
   - \( \sum y^2 \) is the sum of the squares of all the \( y \)-values

2. **Find \( R^2 \)** by squaring the correlation coefficient \( r \).

Let’s calculate this for your data:

**Price in Dollars (x):** 21, 23, 29, 43, 50  
**Number of Bids (y):** 1, 2, 4, 5, 7

I’ll now perform the calculations. numpy as np

# Given data
prices = np.array([21, 23, 29, 43, 50])
bids = np.array([1, 2, 4, 5, 7])

# Calculate sums required for correlation coefficient formula
n = len(prices)
sum_x = np.sum(prices)
sum_y = np.sum(bids)
sum_x2 = np.sum(prices**2)
sum_y2 = np.sum(bids**2)
sum_xy = np.sum(prices * bids)

# Calculate the correlation coefficient (r)
r = (n * sum_xy - sum_x * sum_y) / np.sqrt((n * sum_x2 - sum_x**2) * (n * sum_y2 - sum_y**2))

# Calculate the coefficient of determination (R^2)
r_squared = r**2

r_squaredThe coefficient of determination (\( R^2 \)) for the data is approximately **0.929** (rounded to three decimal places).

This means that about 92.9% of the variance in the number of bids can be explained by the list price of the items.

Would you like further details on how this result is interpreted, or do you have any other questions? Here are some related questions you could explore:

1. What does a correlation coefficient near 1 or -1 signify?
2. How do outliers affect the correlation and \( R^2 \)?
3. What assumptions are made in linear regression?
4. How can we calculate the equation of the regression line from this data?
5. How can we test the statistical significance of the correlation coefficient?

**Tip:** Always check the residuals (differences between observed and predicted values) to assess the appropriateness of your regression model.

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. Step 6 of 6 : Find the value of the coefficient of determination. Round your answer to three decimal places.

Explore the concept of the coefficient of determination (R^2) in linear regression to predict the number of bids received for auction items. This example uses auction data, including item prices and bids, to explain how to calculate R^2 and interpret the results. Perfect for high school students studying statistics.

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