Problem 1 [20 points = 4 + 8 + 4 + 4]
Car sales analysts conducted a study that involved variables X and Y, where X is the age of a car (in years) and Y (in \$K = \$1,000) represents its book value. A sample contained $n = 120$ cars (of the same type) selected for this study. The sample-based summaries are given below.
(i) Means are: $\bar{X} = 6.5$ years and $\bar{Y} = \$13.68K$
(ii) Standard deviations are: $S[X] = 2.5$ years and $S[Y] = \$5K$
(iii) The covariance between X and Y was found as $S[X, Y] = -8$
1. Find the correlation between X and Y
2. Derive the slope and intercept for a linear model, $\hat{Y} = b_0 + b_1 \cdot X$.
3. Sam has the 7.5-year old car ($X = 7.5$) and the trade-in value offered by the dealership was $Y = \$12.5K$. Using the model, determine whether this observation, ($X = 7.5$, $Y = \$12.5K$) would be ABOVE or BELOW the fitted line.
4. Teresa has a 5-year old car ($X = 5$) and she was offered the trade-in value $Y = \$15.4K$. Using the model, determine whether the observation, ($X = 5$, $Y = \$15.4K$), would be ABOVE or BELOW the fitted line.
Solution
