To find the coefficient of determination (denoted as $R^2$), we need to first calculate the correlation coefficient $r$, which can then be squared to obtain $R^2$. Let's go through the process step by step.

The correlation coefficient $r$ is calculated as:

[ 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:

• $x$ is the list price
• $y$ is the number of bids
• $n$ is the number of data points (in this case, 5)

We will start by calculating the following values from the given data:

Price ($x$)	Number of Bids ($y$)	$x^2$	$y^2$	$xy$
20	3	400	9	60
39	5	1521	25	195
43	8	1849	64	344
44	9	1936	81	396
47	10	2209	100	470

Now, let's calculate the necessary sums:

• $\sum x = 20 + 39 + 43 + 44 + 47 = 193$
• $\sum y = 3 + 5 + 8 + 9 + 10 = 35$
• $\sum x^2 = 400 + 1521 + 1849 + 1936 + 2209 = 7915$
• $\sum y^2 = 9 + 25 + 64 + 81 + 100 = 279$
• $\sum xy = 60 + 195 + 344 + 396 + 470 = 1465$
• $n = 5$

Now we can substitute these values into the formula for $r$:

[ r = \frac{5(1465) - (193)(35)}{\sqrt{[5(7915) - (193)^2][5(279) - (35)^2]}} ]

Let's calculate this step by step:

1. Numerator: $5(1465) = 7325, \quad 193(35) = 6755$ $\text{Numerator} = 7325 - 6755 = 570$

2. Denominator: $5(7915) = 39575, \quad 193^2 = 37249 \quad \Rightarrow \quad 39575 - 37249 = 2326$ $5(279) = 1395, \quad 35^2 = 1225 \quad \Rightarrow \quad 1395 - 1225 = 170$ $\text{Denominator} = \sqrt{2326 \times 170} = \sqrt{395420} \approx 629.77$

3. Calculating $r$: $r = \frac{570}{629.77} \approx 0.906$

The coefficient of determination $R^2$ is simply the square of $r$:

$R^2 = (0.906)^2 \approx 0.821$

Final Answer:

The coefficient of determination $R^2$ is approximately 0.821 (rounded to three decimal places).

Would you like a more detailed explanation or any further steps? Here's a list of related questions to explore:

1. How does the coefficient of determination help us evaluate the regression model?
2. What does a high $R^2$ value suggest about the relationship between variables?
3. How do we interpret the slope and y-intercept in the regression line equation?
4. What might it mean if the correlation coefficient is not statistically significant?
5. How would we check for statistical significance of the correlation coefficient?

Tip: The closer $R^2$ is to 1, the stronger the linear relationship between the variables.