To find the estimated slope [tex]\( b_1 \)[/tex] of the regression line [tex]\( \hat{y} = b_0 + b_1 x \)[/tex], we need to analyze the relationship between the prices and the number of bids. Here, we are using the method of least squares linear regression.
Given data:
- List Price (x) in Dollars: [22, 25, 32, 33, 38]
- Number of Bids (y): [3, 5, 6, 9, 10]
The formula for the estimated slope [tex]\( b_1 \)[/tex] in a simple linear regression model is:
[tex]\[ b_1 = \frac{n\sum (xy) - \sum x \sum y}{n\sum (x^2) - (\sum x)^2} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points.
- [tex]\( \sum (xy) \)[/tex] is the sum of the product of paired scores.
- [tex]\( \sum x \)[/tex] is the sum of the x-values.
- [tex]\( \sum y \)[/tex] is the sum of the y-values.
- [tex]\( \sum (x^2) \)[/tex] is the sum of the squares of the x-values.
However, through calculations, the estimated slope is found to be:
[tex]\[ b_1 \approx 0.422 \][/tex]
This value of 0.422 indicates that for each additional dollar increase in the price, the number of bids is expected to increase by approximately 0.422, all else being equal.
