Question

A survey conducted by a research team was to investigate how the education level, tenurein current employment, and age are related to annual income. A sample of 20 employees is selected and the data is given below.
Education (no. of years)	length of tenure in current employment in years	Age in years	Annual income dollars
17	8	40	124000
12	12	41	30000
20	9	44	193000
14	4	42	88000
12	1	19	27000
14	9	28	43000
12	8	43	96000
18	10	37	110000
16	12	36	88000
11	7	39	36000
16	14	36	81000
12	4	22	38000
16	17	45	140000
13	7	42	11000
11	6	18	21000
20	4	40	151000
19	7	35	124000
16	12	38	48000
12	2	19	26000
10	6	44	124000
a.using the excel out provided, please answer the following question. What is the estimated regression equation?
b. please verify that their model is significant at the 0.05 level of significance
c. is each independent variable significant? Conduct t-test using p-value
d.what is the meaning of r square
e. interpret the estimated coefficient (slopes)
f.what is your recommended regression model (equation)
g.estimate annual income for a 35-old individual with 20 years of education and 5 years of tenure.

Answer 1

a.

Using Data analysis package in Excel, we run regression on the above data, we get the output as :

So, Estimated regression equation is :

Y = - 143481.1924 + 10011.92124 X₁ - 2193.883764 X₂ + 2689.240517 X₃

Here, Y = Annual income dollars

X₁ = Education (no. of years)

X₂ = length of tenure in current employment in years

X₃ = Age in years

b. We can see from the ANOVA table that Significance value F = 0.000393507 which is also the P-value.

Since P-value < 0.05 ( level of significance) , we conclude that the model is significant at the 0.05 level of significance.

c.

Null hypothesis H_o : Indepedent variable is not significant

Alternative hypothesis H₁: Indepedent variable is significant

Now we consider the P- value for each variable.

For  X₁ = Education (no. of years), P-value =  0.001287795 < 0.05, we reject Ho and conclude that the independent variable "Education " is significant at 0.05 level of significance.

For  X₂ = length of tenure in current employment in years , P-value = 0.324637926 > 0.05, we do not reject Ho and conclude that the independent variable "length of tenure in current employment" is not significant at 0.05 level of significance.

For  X₃ = Age in years, P-value =  0.014939599 < 0.05, we reject Ho and conclude that the independent variable "Age" is significant at 0.05 level of significance.

d.

R square is the proportion of the variance in the dependent variable that is predictable from the independent variable.

In our case, R-square = 0.670117089 = 67 % (approx)

Hence, 67% of the variance in the dependent variable that is predictable from the conisdered independent variable.

e.

We have , Estimated regression equation :

Y = - 143481.1924 + 10011.92124 X₁ - 2193.883764 X₂ + 2689.240517 X₃

Coefficient of X₁ = 10011.92124 which means that with a unit change in the value of X₁ there is an average increase of 10011.92124 in the value of Y.

Coefficient of X₂ = - 2193.883764 which means that with a unit change in the value of X₂ there is an average decrease of 2193.883764 in the value of Y.

Coefficient of X₃ = 2689.240517 which means that with a unit change in the value of X₃ there is an average increase of 2689.240517 in the value of Y.

f.

Recommended regression model would be to just remove the variable X₂ = length of tenure in current employment in years from ourr model since it is coming out to be insignificant.

When we again run the model after removing X₂ we get th equation as :

Y = -138671.6837 + 9588.89332 X₁ + 2234.556096 X₃

g.

We use the estimated regression equation in the part (a) :

Y = - 143481.1924 + 10011.92124 X₁ - 2193.883764 X₂ + 2689.240517 X₃

We are given, X₁ = 20 , X₂ = 5 , X₃ = 35

Y = - 143481.1924 + 10011.92124 * 20 - 2193.883764 * 5 + 2689.240517 * 35 = 139911.2

Estimated Annual income in dollars = 139911.2