#5 Please solve with all steps and answer all parts ! 25,000 38,000 28,000 35,00

Question

#5

Please solve with all steps and answer all parts !

25,000 38,000 28,000 35,000 30 000 32,000 Salary Fit these data with a regression model of the form y x r. Plot the observed salaries and the predicted salaries. Is the regres sion fit reasonably good? (b) Repeat the calculations in part (a) by first shifting the x-values to make the average x-value be 0 (see equations (21) 5. Consider the following relationship between the height of a student's mother and the number of F's the student gets at Podunk U.

Explanation / Answer

Back-up Theory

X = mother’s height; Y = Number of F’s

Then, Mean X = Xbar = (1/n)sum of xi over I = 1, 2, …., n; ……………….(1)

Variance of X, V(X) = (1/n)Sxx where Sxx

= sum of (xi – Xbar)2 over i = 1, 2, …., n ………………………………..(2)

Standard Deviation of X = SDX = sq.rt of V(X). ……………………………(3)

Similarly, Mean Y = Ybar =(1/n)sum of yiover i= 1, 2, …., n;…………….(4)

Variance of Y, V(Y) = (1/n)Syy where Syy

= sum of (yi – Ybar)2 over i = 1, 2, …., n ………………………………………………(5)

Standard Deviation of Y = SDY = sq.rt of V(Y). ………………………….……………….(6)

Covariance of X and Y, Cov(X, Y)

= (1/n)Sxy where Sxy = sum of {(xi – Xbar)(yi – Ybar)} over i = 1, 2, …., n………(7)

Estimated Regression of Y on X is given by: Ycap= qX + r, where

q = Cov(X, Y)/V(X) = Sxy/Sxx and r = Ybar – q.Xbar..…………………….(8)

Now, to work out solution,

Using Excel Function, the computations yield the following:

n = 8; Xbar = 62.625;     Ybar = 4.25;    Sxy = 12.75    Sxx = 73.876;

q = 0.1726, r = - 6.5584  

Part (a)

So, the estimated regression is: ycap = 0.1726x – 6.5584 ANSWER

Part (b)

All the above values after deleting the values for Student E are given below.

n = 7; Xbar = 63;     Ybar = 3.2857;    Sxy = 33    Sxx = 66;

q = 0.5, r = - 28.2143  

So, the estimated regression is: ycap = 0.5x – 28.2143 ANSWER

Comment: The ‘No of F’s’ for student E is very high compared to other students’ – a real outlier. This has actually marred the real regression equation estimate. After deleting this, the data is fairly homogenous and hence the estimated regression is likely to more reliable.

Part (c)

Shifting x-values to make the average zero is equivalent saying ‘subtract the average (63) from all x values.

All the above values after doing this operation E are given below.

n = 7; Xbar = 0;     Ybar = 3.2857;    Sxy = 33    Sxx = 66;

q = 0.5, r = 3.2857  

So, the estimated regression is: ycap = 0.5x + 3.2857 ANSWER

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