e table below gives the list price and the number of bids recelved for five randomly selected items sold through online auctions. Using this data, consider the equation the regression line, $\hat{y}=b_{0}+b_{1} x$, for predicting the number of bids an item will recelve based on the list price. Keep in mind, the correlation coefficient may or may t be statistically significant for the data given. Remember, in practice, It would not be appropriate to use the regression line to make a prediction if the correlation efficient is not statistically significant.
\begin{tabular}{|l|c|c|c|c|c|}
\hline Price in Dollars & 30 & 32 & 43 & 45 & 50 \\
\hline Number of Bids & 2 & 4 & 5 & 6 & 7 \\
\hline
\end{tabular}
Table
Copy Data
itep 5 of 6: Determine the value of the dependent varlable $\hat{y}$ at $x=0$.
Tables
Keypad
Keyboard Shortcuts
Previous step answers
$O b_{0}$
$b_{1}$
$x$
Submit Answer
Hawkes Learning

Step 1 ：The problem provides a table of list prices and the number of bids received for five items sold through online auctions. We are asked to find the value of the dependent variable \(\hat{y}\) at \(x=0\). This is essentially asking for the y-intercept of the regression line, which is represented by \(b_{0}\) in the equation \(\hat{y}=b_{0}+b_{1} x\).

Step 2 ：To find this, we need to calculate the slope \(b_{1}\) and the y-intercept \(b_{0}\) of the regression line using the given data. The formulas for these are: \(b_{1} = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\) and \(b_{0} = \bar{y} - b_{1}\bar{x}\), where \(n\) is the number of data points, \(\sum xy\) is the sum of the product of each pair of x and y values, \(\sum x\) and \(\sum y\) are the sums of the x and y values respectively, \(\sum x^2\) is the sum of the squares of the x values, and \(\bar{x}\) and \(\bar{y}\) are the means of the x and y values respectively.

Step 3 ：Using the given data, we find that \(x = [30, 32, 43, 45, 50]\), \(y = [2, 4, 5, 6, 7]\), \(n = 5\), \(\sum x = 200\), \(\sum y = 24\), \(\sum xy = 1023\), \(\sum x^2 = 8298\), \(\bar{x} = 40.0\), and \(\bar{y} = 4.8\).

Step 4 ：Substituting these values into the formulas, we find that \(b_{1} = 0.21140939597315436\) and \(b_{0} = -3.6563758389261745\).

Step 5 ：Final Answer: The value of the dependent variable \(\hat{y}\) at \(x=0\) is \(\boxed{-3.6563758389261745}\). Therefore, \(b_{0} = \boxed{-3.6563758389261745}\).