To find the estimated value of $y$ when $x = 27$, we need to follow these steps:

1. Calculate the regression line $\hat{y} = b_0 + b_1 x$, where:
   • $b_1$ is the slope of the regression line, and
   • $b_0$ is the y-intercept.

Step 1: Calculate the slope $b_1$ and intercept $b_0$

We use the formulas for the slope and intercept of the regression line:

$b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$ $b_0 = \frac{\sum y - b_1 \sum x}{n}$

Where:

• $n$ is the number of data points,
• $\sum x$ is the sum of all the $x$-values (prices),
• $\sum y$ is the sum of all the $y$-values (number of bids),
• $\sum xy$ is the sum of the product of corresponding $x$- and $y$-values,
• $\sum x^2$ is the sum of the squares of the $x$-values.

Step 2: Set up the data and compute the necessary values

Given data:

• $x$ (prices): 21, 27, 31, 43, 49
• $y$ (number of bids): 4, 6, 7, 8, 9

We will compute the sums:

• $\sum x = 21 + 27 + 31 + 43 + 49 = 171$
• $\sum y = 4 + 6 + 7 + 8 + 9 = 34$
• $\sum x^2 = 21^2 + 27^2 + 31^2 + 43^2 + 49^2 = 441 + 729 + 961 + 1849 + 2401 = 5381$
• $\sum xy = (21 \times 4) + (27 \times 6) + (31 \times 7) + (43 \times 8) + (49 \times 9) = 84 + 162 + 217 + 344 + 441 = 1248$

Now, we can calculate the slope $b_1$ and intercept $b_0$.

Slope $b_1$:

$b_1 = \frac{5 \times 1248 - 171 \times 34}{5 \times 5381 - 171^2}$

$b_1 = \frac{6240 - 5814}{26905 - 29241}$

$b_1 = \frac{426}{-234}$

$b_1 = -1.82 \quad (\text{approximately})$

Intercept $b_0$:

$b_0 = \frac{34 - (-1.82) \times 171}{5}$

$b_0 = \frac{34 + 310.62}{5}$

$b_0 = \frac{344.62}{5} = 68.924$

Step 3: Find the estimated value of $y$ when $x = 27$

Now that we have $b_0 = 68.924$ and $b_1 = -1.82$, we can substitute $x = 27$ into the regression equation:

$\hat{y} = 68.924 + (-1.82)(27)$

$\hat{y} = 68.924 - 49.14$

$\hat{y} = 19.784$

Thus, the estimated value of $y$ when $x = 27$ is 19.784.

Would you like me to explain any part of the process, or is there anything else you'd like to explore? Here are some related questions for further practice:

1. How would you interpret the slope of the regression line in this context?
2. What would happen to the predicted number of bids if the list price were increased by $10?
3. How do you check if the correlation coefficient is statistically significant?
4. How would you calculate the residuals for each data point?
5. What does the $R^2$ value tell you about the fit of the regression line?

Tip: It's always a good practice to check the residuals after fitting a regression model. They can tell you how well the model fits the data and if any patterns are left unexplained.