To find the estimated slope $b_1$ of the regression line $\hat{y} = b_0 + b_1 x$, we can use the formula for the slope of the regression line:

$b_1 = \frac{n \sum{xy} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}$

Where:

• $n$ is the number of data points (in this case, 5).
• $x$ is the list price, and $y$ is the number of bids.
• $\sum{x}$, $\sum{y}$, $\sum{xy}$, and $\sum{x^2}$ are the sums of the values of $x$, $y$, the product of $x$ and $y$, and the square of $x$, respectively.

Data from the table:

Price in Dollars $x$	Number of Bids $y$
20	3
39	5
43	8
44	9
47	10

Step-by-step calculation:

1. Sum of $x$: $\sum{x} = 20 + 39 + 43 + 44 + 47 = 193$

2. Sum of $y$: $\sum{y} = 3 + 5 + 8 + 9 + 10 = 35$

3. Sum of $x^2$ (squares of prices): $\sum{x^2} = 20^2 + 39^2 + 43^2 + 44^2 + 47^2 = 400 + 1521 + 1849 + 1936 + 2209 = 7915$

4. Sum of $xy$ (product of price and number of bids): $\sum{xy} = (20 \times 3) + (39 \times 5) + (43 \times 8) + (44 \times 9) + (47 \times 10) = 60 + 195 + 344 + 396 + 470 = 1465$

Now, applying these sums to the formula for $b_1$:

$b_1 = \frac{5 \times 1465 - 193 \times 35}{5 \times 7915 - (193)^2}$ $b_1 = \frac{7325 - 6755}{39575 - 37249}$ $b_1 = \frac{570}{2326}$ $b_1 \approx 0.245$

The estimated slope $b_1$ is approximately 0.245 (rounded to three decimal places).

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Do you want to continue with the remaining steps? If you have any questions or need further clarification, feel free to ask!

Here are some related questions to deepen the understanding:

1. How would the regression line change if the list prices were doubled?
2. What is the interpretation of the slope $b_1$ in the context of this problem?
3. How would you calculate the estimated y-intercept $b_0$ using the regression formula?
4. How can you assess whether the correlation between price and bids is statistically significant?
5. What impact would outliers in the data have on the regression line?

Tip: Double-check your sums and arithmetic for accuracy when calculating regression coefficients.