In: Statistics and Probability
A sample of 6 homes sold last week was selected. Is there a relationship between ? = the home size (reported in hundreds of square feet) and ? = the selling price (reported in $ thousands)? ? (sq.ft) 14 13 11 13 8 10 ? ($) 140 105 120 118 78 102
STEP 1: Create a scatter diagram. USE GRAPH PAPER! (provided) Using only the scatter diagram (NO CALCULATIONS). Would you estimate the correlation coefficient to be positive, close to zero or negative? Explain your answer.
STEP 2: For the given data compute each of the following ??, ??, ?? 2 , ?? 2 , ???, ?̅ ??? ?̅, ? ? ? 2 ? 2 ?? ∑ = ?̅ = ?̅ =
STEP 3: Compute the sample correlation coefficient ?. What is the interpretation of your correlation coefficient?
STEP 5: Compute the slope ? and y-intercept, a, of the least squares line; write out the equation for the least squares line. (SHOW YOUR WORK!)
STEP 6: Solve your regression equation for the lowest and highest values in your sample data and plot on your graph. Draw the least squares line on your scatter diagram. (Your least squares line must be done on the graph paper provided. Do not submit an excel printout for your graph.)
STEP 7: Compute the coefficient of determination. Provide an explanation of the meaning of the coefficient of determination in the context of this problem. • Suppose a house recently went on the market with a size of 12 hundred square feet. What does the least squares line predict as the selling price for this house? (be sure to label your answer correctly
Step 1
Scatter diagram:
From scatter diagram see that there is positive correlation between X and Y. Because as value of X increases value of Y also increases and voice a versa.
Step.2
X | Y | X2 | Y2 | XY |
14 | 140 | 196 | 19600 | 1960 |
13 | 105 | 169 | 11025 | 1365 |
11 | 120 | 121 | 14400 | 1320 |
13 | 118 | 169 | 13924 | 1534 |
8 | 78 | 64 | 6084 | 624 |
10 | 102 | 100 | 10404 | 1020 |
69 | 663 | 4761 | 439569 | 45747 |
138 | 1326 | 5580 | 515006 | 53570 |
Step.3
Sample correlation coefficient =
Step.5
Slope:
Intercept:
Least square regression line is:
Y = 0.411 + 9.592 X
Step.6
Smallest X = 8 then estimated Y is
Y = 0.411 + 9.592 (8) = 77.147
Largest Y= 69 then
estimated Y is
Y = 0.411 + 9.592 (69) = 662.259