TDSM 2.11

Let $$X$$ be the annual salaries of high school graduates

$$Y$$ be the annual salaries of high school graduates

$$n$$ be the number of job positions

a) For each possible job title, the college graduates always made 5,000 dollars more than high school grads

$$\Rightarrow \bar{Y} = \bar{X} + 5000$$ and $$\forall i (1 \leq i \leq n): Y_i = X_i + 5000$$

Correlation efficient of $$X$$ and $$Y$$ :

$$\tau = \frac{\sum_{i = 1}^{n}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \sqrt{\sum_{i = 1}^{n}(Y_i - \bar{Y})^2}} = \frac{\sum_{i = 1}^{n}(X_i - \bar{X})(X_i + 5000 - (\bar{X} + 5000))}{\sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \sqrt{\sum_{i = 1}^{n}(X_i + 5000 - (\bar{X} + 5000))^2}} = \frac{\sum_{i = 1}^{n}(X_i - \bar{X})^2} {\left( \sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \right)^2} = 1$$

b) For each possible job title, the college graduates always made 25% more than high school grads

$$\Rightarrow$$

c) For each possible job title, the college graduates always made 15% less than high school grads

$$\Rightarrow$$