Difference between revisions of "TDSM 2.11"
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Let <math>X</math> be the annual salaries of high school graduates | Let <math>X</math> be the annual salaries of high school graduates | ||
| − | <math>Y</math> be the annual salaries of | + | <math>Y</math> be the annual salaries of college graduates |
<math>n</math> be the number of job positions | <math>n</math> be the number of job positions | ||
| Line 9: | Line 9: | ||
<math>\Rightarrow \bar{Y} = \bar{X} + 5000</math> and <math> \forall i (1 \leq i \leq n): Y_i = X_i + 5000 </math> | <math>\Rightarrow \bar{Y} = \bar{X} + 5000</math> and <math> \forall i (1 \leq i \leq n): Y_i = X_i + 5000 </math> | ||
| − | Correlation | + | Correlation coefficient of <math>X</math> and <math>Y</math>: |
<math> | <math> | ||
| Line 18: | Line 18: | ||
</math> | </math> | ||
| + | b) For each possible job title, the college graduates always made 25% more than high school grads | ||
| + | |||
| + | <math>\Rightarrow \bar{Y} = 1.25 \bar{X}</math> and <math> \forall i (1 \leq i \leq n): Y_i = 1.25 X_i </math> | ||
| − | + | Correlation coefficient of <math>X</math> and <math>Y</math>: | |
| − | <math>\ | + | <math> |
| + | \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}) \cdot 1.25 (X_i - \bar{X})}{\sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \sqrt{ \sum_{i = 1}^{n}1.25^2(X_i - \bar{X})^2}} | ||
| + | = \frac{1.25 \sum_{i = 1}^{n}(X_i - \bar{X})^2} {1.25 \left( \sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \right)^2} | ||
| + | = 1 | ||
| + | </math> | ||
c) For each possible job title, the college graduates always made 15% less than high school grads | c) For each possible job title, the college graduates always made 15% less than high school grads | ||
| − | <math>\Rightarrow</math> | + | |
| + | <math>\Rightarrow \bar{Y} = 0.85 \bar{X}</math> and <math> \forall i (1 \leq i \leq n): Y_i = 0.85 X_i </math> | ||
| + | |||
| + | Correlation coefficient of <math>X</math> and <math>Y</math>: | ||
| + | |||
| + | <math> | ||
| + | \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}) \cdot 0.85 (X_i - \bar{X})}{\sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \sqrt{ \sum_{i = 1}^{n}0.85^2(X_i - \bar{X})^2}} | ||
| + | = \frac{0.85 \sum_{i = 1}^{n}(X_i - \bar{X})^2} {0.85 \left( \sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \right)^2} | ||
| + | = 1 | ||
| + | </math> | ||
Latest revision as of 15:30, 12 December 2017
Let [math]X[/math] be the annual salaries of high school graduates
[math]Y[/math] be the annual salaries of college graduates
[math]n[/math] 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
[math]\Rightarrow \bar{Y} = \bar{X} + 5000[/math] and [math] \forall i (1 \leq i \leq n): Y_i = X_i + 5000 [/math]
Correlation coefficient of [math]X[/math] and [math]Y[/math]:
[math] \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 [/math]
b) For each possible job title, the college graduates always made 25% more than high school grads
[math]\Rightarrow \bar{Y} = 1.25 \bar{X}[/math] and [math] \forall i (1 \leq i \leq n): Y_i = 1.25 X_i [/math]
Correlation coefficient of [math]X[/math] and [math]Y[/math]:
[math] \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}) \cdot 1.25 (X_i - \bar{X})}{\sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \sqrt{ \sum_{i = 1}^{n}1.25^2(X_i - \bar{X})^2}} = \frac{1.25 \sum_{i = 1}^{n}(X_i - \bar{X})^2} {1.25 \left( \sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \right)^2} = 1 [/math]
c) For each possible job title, the college graduates always made 15% less than high school grads
[math]\Rightarrow \bar{Y} = 0.85 \bar{X}[/math] and [math] \forall i (1 \leq i \leq n): Y_i = 0.85 X_i [/math]
Correlation coefficient of [math]X[/math] and [math]Y[/math]:
[math] \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}) \cdot 0.85 (X_i - \bar{X})}{\sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \sqrt{ \sum_{i = 1}^{n}0.85^2(X_i - \bar{X})^2}} = \frac{0.85 \sum_{i = 1}^{n}(X_i - \bar{X})^2} {0.85 \left( \sqrt{\sum_{i = 1}^{n}(X_i - \bar{X})^2} \right)^2} = 1 [/math]
