Question

4. A story in the Wall Street Journal reveals that “Tall Workers Earn More Money”. Ron, the president of a company, wants to know whether the relationship between salary and height exists in his own company, and obtains the following data from the human resources department.

a. Calculate the correlation between salary and height, and describe the relationship between these two variables.

b. How much of the variability in salary can be accounted for by height?

c. Is there a significant correlation between salary and height?

Show all your work. Full marks will not be given unless all steps and all calculations are shown. For all hypothesis tests, include all steps. Remember to provide concluding statements in the proper format.

Employee	Salary (X) (in $1000s of dollars)	Height (Y) (in inches)
Max	45	72
Jacob	38	70
Jesse	39	74
Jennifer	33	60
Jeremy	40	63
Brian	36	68
Barbara	42	67
Benny	35	64
Rhonda	47	77
Kelly	45	69
Miriam	37	67
Gloria	34	66
Shirley	42	70
Eden	46	73

Answer 1

a)

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
45	72	25.72	11.76	17.39
38	70	3.72	2.04	-2.76
39	74	0.86	29.47	-5.04
33	60	48.01	73.47	59.39
40	63	0.01	31.04	-0.40
36	68	15.43	0.33	2.24
42	67	4.29	2.46939	-3.2551
35	64	24.29	20.89796	22.531
47	77	50.01	71.04	59.60
45	69	25.72	0.18	2.17
37	67	8.58	2.47	4.60
34	66	35.15	6.61	15.24
42	70	4.29	2.04	2.96
46	73	36.86	19.61	26.89

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	559.00	960.00	282.93	273.43	201.57
mean	39.93	68.57	SSxx	SSyy	SSxy

correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.7247

b) R² =    (Sxy)²/(Sx.Sy) =    0.5252 or 52.52%

c)

Ho:   ρ = 0  
Ha:   ρ ╪ 0  
n=   14  
alpha,α =    0.05  
correlation , r=   0.7247  
t-test statistic = r*√(n-2)/√(1-r²) =        3.643
DF=n-2 =   12  
p-value =    0.0034  
Decison:   p value < α , So, Reject Ho  

there is a significant relationship