To find the estimated value of $y$ when $x = 27$, we need to determine the regression equation:

$\hat{y} = b_0 + b_1x$

Step 1: Compute the Slope ($b_1$)

The formula for the slope is:

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

Given data:

Price ($x$)	Number of Bids ($y$)	$xy$	$x^2$
21	4	84	441
27	6	162	729
31	7	217	961
43	8	344	1849
49	9	441	2401

Now, compute the required summations:

$\sum x = 21 + 27 + 31 + 43 + 49 = 171$

$\sum y = 4 + 6 + 7 + 8 + 9 = 34$

$\sum xy = 84 + 162 + 217 + 344 + 441 = 1248$

$\sum x^2 = 441 + 729 + 961 + 1849 + 2401 = 6381$

$n = 5$

$b_1 = \frac{(5)(1248) - (171)(34)}{(5)(6381) - (171)^2}$

$b_1 = \frac{6240 - 5814}{31905 - 29241} = \frac{426}{2664} \approx 0.160$

Step 2: Compute the Intercept ($b_0$)

The intercept is given by:

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

where

$\bar{x} = \frac{\sum x}{n} = \frac{171}{5} = 34.2$

$\bar{y} = \frac{\sum y}{n} = \frac{34}{5} = 6.8$

$b_0 = 6.8 - (0.160)(34.2)$

$b_0 = 6.8 - 5.472 = 1.328$

Step 3: Predict $y$ when $x = 27$

$\hat{y} = 1.328 + (0.160)(27)$

$\hat{y} = 1.328 + 4.32 = 5.648$

Final Answer: $\hat{y} \approx 5.648$

Would you like me to verify significance of the correlation coefficient?

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Related Questions:

1. How do we interpret the correlation coefficient in this context?
2. What is the coefficient of determination ($R^2$) and how does it relate to the regression line?
3. How can we test if the regression model is statistically significant?
4. What are the assumptions of linear regression, and do they apply here?
5. How do outliers impact the accuracy of a regression model?

Tip:

Always check the significance of the correlation coefficient before making predictions with a regression model. If it is not statistically significant, predictions may not be reliable.