To find the coefficient of determination ($R^2$), we first need to calculate the correlation coefficient ($r$), and then square it to obtain $R^2$. The coefficient of determination indicates how well the regression line explains the variability in the number of bids based on the list price.

Given Data:

• Prices ($x$): 23, 34, 44, 46, 50
• Number of bids ($y$): 1, 2, 4, 9, 10

Steps to Calculate the Coefficient of Determination:

1. Calculate the Mean of $x$ and $y$: $\bar{x} = \frac{23 + 34 + 44 + 46 + 50}{5} = \frac{197}{5} = 39.4$ $\bar{y} = \frac{1 + 2 + 4 + 9 + 10}{5} = \frac{26}{5} = 5.2$

2. Calculate the Covariance of $x$ and $y$: $\text{Cov}(x, y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n}$ Where $n = 5$ is the number of data points.

3. Calculate the Standard Deviations of $x$ and $y$: $\sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$ $\sigma_y = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n}}$

4. Find the Correlation Coefficient $r$: $r = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y}$

5. Calculate the Coefficient of Determination $R^2$: $R^2 = r^2$

Let me now calculate each of these steps to find the $R^2$ value. numpy as np

Given data

x = np.array([23, 34, 44, 46, 50]) y = np.array([1, 2, 4, 9, 10])

Step 1: Calculate the means of x and y

x_bar = np.mean(x) y_bar = np.mean(y)

Step 2: Calculate Covariance

cov_xy = np.mean((x - x_bar) * (y - y_bar))

Step 3: Calculate Standard Deviations of x and y

sigma_x = np.std(x, ddof=0) sigma_y = np.std(y, ddof=0)

Step 4: Calculate the correlation coefficient

r = cov_xy / (sigma_x * sigma_y)

Step 5: Calculate the coefficient of determination

R_squared = r**2

R_squaredThe value of the coefficient of determination ($R^2$) is approximately 0.773.

This means that about 77.3% of the variability in the number of bids can be explained by the list price according to the regression model.

Would you like any further details on the calculation or the interpretation of the result?

Here are 5 related questions you might find helpful:

1. What does a high value of $R^2$ imply in terms of prediction accuracy?
2. How do we interpret the correlation coefficient $r$ in this context?
3. What are the conditions for using linear regression in predicting outcomes like this?
4. How do you calculate the regression line equation for this data?
5. How can we test the statistical significance of the correlation coefficient?

Tip: When interpreting $R^2$, remember that a higher value indicates that the regression model explains a greater proportion of the variance in the dependent variable.