To predict the size of a house costing 450 thousand dollars, we can use linear regression. Here's a step-by-step approach:

1. Set up your data:

   • Size (X): 1200, 1339, 1382, 2351, 2369, 3600
   • Price (Y): 320, 430, 389, 635, 689, 750

2. Calculate the necessary statistics:

   • Means: $\bar{X}$ and $\bar{Y}$
   • Slope (b) = $\frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}}$
   • Intercept (a) = $\bar{Y} - b\bar{X}$

3. Substitute to predict:

   • Regression equation: $Y = a + bX$
   • For 450: $450 = a + bX$
   • Solve for X.

Let's calculate:

• Mean of X ($\bar{X}$): $(1200 + 1339 + 1382 + 2351 + 2369 + 3600)/6 \approx 2040.17$

• Mean of Y ($\bar{Y}$): $(320 + 430 + 389 + 635 + 689 + 750)/6 \approx 535.5$

• Applying the formula for the slope (b):

$b = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}}$

Calculating the sums:

$\sum{(X_i - \bar{X})(Y_i - \bar{Y})} \approx 4,548,144$ $\sum{(X_i - \bar{X})^2} \approx 2,130,845$

Thus,

$b \approx \frac{4,548,144}{2,130,845} \approx 2.13$

• Intercept (a):

$a = \bar{Y} - b\bar{X} \approx 535.5 - 2.13(2040.17) \approx -1875.86$

• Regression equation:

$Y = -1875.86 + 2.13X$

Now, to find the size (X) for a price (Y) of 450:

$450 = -1875.86 + 2.13X$ $2.13X = 450 + 1875.86$ $X \approx \frac{2325.86}{2.13} \approx 1091.95$

Thus, the predicted size of a house costing 450 thousand dollars is approximately 1092 square feet.