Answer to Question #280915 in Statistics and Probability for J.K

Question

1.     A nation job placement company is interested in developing a model that might be used to explain the variation in starting salaries for college graduates based on the college GPA.  The following data were collected through a random sample of the clients with which this company has been associated.

 

GPA

Starting Salary

3.20

OMR35,000

3.40

OMR29,500

2.90

OMR30,000

3.60

OMR36,400

2.80

OMR31,500

2.50

OMR29,000

3.00

OMR33,200

3.60

OMR37,600

2.90

OMR32,000

3.50

OMR36,000

 

 

Based on this sample information, determine the least squares regression model. Also, develop a scatter plot of the data and locate the regression line on the scatter plot.   

Answer 1

GPA Score, $X$

$$3.20,3.40,2.90,3.60,2.80,$$

$$2.50,3.00,3.60,2.90,3.50$$

Starting Salary (RM), $Y$

$$3500,2950,3000,3640,3150,$$

$$2900,3320,3760,3200,3600$$

$$\bar{X}=\dfrac{1}{n}\sum _iX_i=\dfrac{31.4}{10}=3.14$$

$$\bar{Y}=\dfrac{1}{n}\sum _iY_i=\dfrac{33020}{10}=3302$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _iX_i)^2$$

$$=99.88-\dfrac{(31.4)^2}{10}=1.284$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _iY_i)^2$$

$$=109894600-\dfrac{(33020)^2}{10}=862560$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _iX_i)(\sum _iY_i)$$

$$=104480-\dfrac{31.4(33020)}{10}=797.2$$

$$\beta_1=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{797.2}{1.284}=620.8723$$

$$\beta_0=\bar{Y}-\beta_1\cdot\bar{X}=3302-\dfrac{797.2}{1.284}(3.14)$$

$$=1352.4611$$

The regression equation is:

$$Y=1352.4611+620.8723X$$

b)

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}$$

$$=\dfrac{797.2}{\sqrt{1.284}\sqrt{862560}}=0.7575$$

$$r>0.7$$

Positive strong correlation.