To find the estimated y-intercept $b_0$ for the regression line $\hat{y} = b_0 + b_1x$, we first need to compute the slope $b_1$ and use the formula for the y-intercept:

$b_0 = \bar{y} - b_1 \bar{x}$

Where:

• $\bar{y}$ is the mean of the number of bids.
• $\bar{x}$ is the mean of the list prices.
• $b_1$ is the slope of the regression line, which is calculated as:

$b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

Step 1: Compute the means

We have the following data:

List prices ($x$): 29, 33, 34, 36, 46
Number of bids ($y$): 1, 3, 8, 9, 10

First, let's calculate the means of $x$ and $y$.

$\bar{x} = \frac{29 + 33 + 34 + 36 + 46}{5} = \frac{178}{5} = 35.6$ $\bar{y} = \frac{1 + 3 + 8 + 9 + 10}{5} = \frac{31}{5} = 6.2$

Step 2: Calculate $b_1$

Now, we calculate the numerator and denominator for the slope $b_1$.

1. Compute the deviations $(x_i - \bar{x})$ and $(y_i - \bar{y})$, and their products:

$x_i$	$y_i$	$x_i - \bar{x}$	$y_i - \bar{y}$	$(x_i - \bar{x})(y_i - \bar{y})$	$(x_i - \bar{x})^2$
29	1	-6.6	-5.2	34.32	43.56
33	3	-2.6	-3.2	8.32	6.76
34	8	-1.6	1.8	-2.88	2.56
36	9	0.4	2.8	1.12	0.16
46	10	10.4	3.8	39.52	108.16

Now sum the columns for $(x_i - \bar{x})(y_i - \bar{y})$ and $(x_i - \bar{x})^2$:

$\sum (x_i - \bar{x})(y_i - \bar{y}) = 34.32 + 8.32 - 2.88 + 1.12 + 39.52 = 80.4$

$\sum (x_i - \bar{x})^2 = 43.56 + 6.76 + 2.56 + 0.16 + 108.16 = 161.2$

Now, calculate the slope $b_1$:

$b_1 = \frac{80.4}{161.2} \approx 0.499$

Step 3: Calculate the y-intercept $b_0$

Finally, use the formula for $b_0$:

$b_0 = \bar{y} - b_1 \bar{x} = 6.2 - (0.499)(35.6) = 6.2 - 17.74 = -11.54$

Final Answer

The estimated y-intercept $b_0$ is approximately -11.540.

Would you like to proceed with more details, or do you have any further questions on this? Here are some related questions to deepen the understanding:

1. How would you compute the slope and intercept for a larger dataset?
2. What is the interpretation of the regression equation in this context?
3. How do you test the significance of the correlation coefficient?
4. What would happen if the correlation coefficient were zero?
5. How can residuals help assess the quality of the regression model?

Tip: Always check the assumptions of linear regression (linearity, independence, homoscedasticity, and normality of residuals) before making predictions.