To find the estimated slope $b_1$ of the regression line $\hat{y} = b_0 + b_1x$, we use the formula:

$b_1 = \frac{n \sum{x_i y_i} - \sum{x_i} \sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}$

Where:

• $n$ is the number of data points (in this case, $n = 5$),
• $x_i$ are the prices,
• $y_i$ are the number of bids,
• $\sum{x_i}$ is the sum of all the prices,
• $\sum{y_i}$ is the sum of all the number of bids,
• $\sum{x_i^2}$ is the sum of the squares of the prices,
• $\sum{x_i y_i}$ is the sum of the product of the price and the number of bids.

Let's calculate each of these:

Step 1: Compute the necessary sums

1. List of prices $x$: 21, 27, 31, 43, 49
2. List of number of bids $y$: 4, 6, 7, 8, 9

Sums:

• $\sum{x_i} = 21 + 27 + 31 + 43 + 49 = 171$
• $\sum{y_i} = 4 + 6 + 7 + 8 + 9 = 34$
• $\sum{x_i^2} = 21^2 + 27^2 + 31^2 + 43^2 + 49^2 = 441 + 729 + 961 + 1849 + 2401 = 6381$
• $\sum{x_i y_i} = (21 \times 4) + (27 \times 6) + (31 \times 7) + (43 \times 8) + (49 \times 9) = 84 + 162 + 217 + 344 + 441 = 1248$

Now, plug these values into the formula for $b_1$:

$b_1 = \frac{5(1248) - (171)(34)}{5(6381) - (171)^2}$ $b_1 = \frac{6240 - 5814}{31905 - 29241}$ $b_1 = \frac{426}{2664}$ $b_1 \approx 0.160$

Thus, the estimated slope $b_1$ is approximately 0.160.

Would you like help with the next steps, or do you have any questions about this calculation?

Related questions:

1. How can we interpret the slope $b_1$ in the context of the problem?
2. What is the next step to find the intercept $b_0$ of the regression line?
3. How can we check if the correlation coefficient is statistically significant?
4. What does it mean if the correlation coefficient is not statistically significant for this regression line?
5. How does the regression model help us predict the number of bids based on the price?

Tip: Always check if the data you are using falls within the range of the values used to calculate the regression model before making predictions.