Question

You are a manager for the human resource team of a company. You are about to hire a new employee. This new employee is a good fit for the position, and you are about to make an offer. Now it is time to negotiate on the salary. This new employee claims that he has over 20 years of experience and are currently earning $160,000 annually in his previous company. So, he is asking for at least more than $160,000. Before you make the offer, you want to verify what he said is true. You managed to get a simple table of salaries for 10 different positions in his previous company. Another very relevant information is that this new employee that is about to be hired has been a regional manager for two years now in the previous company, and that it usually takes on average four years to jump from being a regional manager to a partner. This new employee about to be hired was halfway between level 6 and level 7 and therefore we can say he was level 6.5. (1) For each of these statements, answer true or false or can’t tell (meaning there is not enough information given)

a. There is a association between X and Y. ______

b. There is a linear relationship between X and Y. _______

c. X and Y are strongly correlated. _______

d. There is a nonlinear relationship between X and Y._____

e. There is a causal relationship between X and Y. ______

(2) Use a regression model to predict if this new employee was bluffing about his salary.

Position	Level	Salary
Business Analyst	1	45000
Junior Consultant	2	50000
Senior Consultant	3	60000
Manager	4	80000
Country Manager	5	110000
Region Manager	6	150000
Partner	7	200000
Senior Partner	8	300000
C-level	9	500000
CEO	10	1000000

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1	45000	20.2500	41820250000.0000	920250.000
2	50000	12.2500	39800250000.0000	698250.000
3	60000	6.2500	35910250000.0000	473750.000
4	80000	2.2500	28730250000.0000	254250.000
5	110000	0.2500	19460250000.0000	69750.000
6	150000	0.2500	9900250000.0000	-49750.000
7	200000	2.2500	2450250000.0000	-74250.000
8	300000	6.2500	2550250000.0000	126250.000
9	500000	12.2500	62750250000.0000	876750.000
10	1000000	20.2500	563250250000.0000	3377250.000

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	55.00	2495000.00	82.50	806622500000.00	6672500.00
mean	5.50	249500.00	SSxx	SSyy	SSxy

Sample size,   n =   10      
here, x̅ = Σx / n=   5.500          
ȳ = Σy/n =   249500.000          
SSxx =    Σ(x-x̅)² =    82.5000      
SSxy=   Σ(x-x̅)(y-ȳ) =   6672500.0      
              
estimated slope , ß1 = SSxy/SSxx =   6672500/82.5=   80878.7879      
intercept,ß0 = y̅-ß1* x̄ =   249500- (80878.7879 )*5.5=   -195333.3333      
              
Regression line is, Ŷ=   -195333.333   + (   80878.788   )*x
              
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    266958787878.7880      
std error ,Se =    √(SSE/(n-2)) =    182674.1593      
              
correlation coefficient ,    r = SSxy/√(SSx.SSy) =   0.81795      
R² =    (SSxy)²/(SSx.SSy) =    0.6690      

Predicted Y at X=   6.5   is          
Ŷ=   -195333.33333   +   80878.78788   *6.5=   330379

a. There is a association between X and Y. ___True___

b. There is a linear relationship between X and Y. False

c. X and Y are strongly correlated. ______False_

d. There is a nonlinear relationship between X and Y.___True__

e. There is a causal relationship between X and Y. ____Can't tell__

Please let me know in case of any doubt.

Thanks in advance!

Please upvote!