Please see the attached file.

1. In order to asses the profitability of additional hours spent on various projects, a manager decided to regress profitability of the project (Profits) vs. the number of hours spent on developing a project (Time). Profits are expressed in thousands of dollars. The results of the regressions are given below:

Regression: Profits
Constant Time
Coefficient 3.35738712 1.8163041
Std error of coef 1.965806 0.3640728
t-ratio 1.7079 4.9888
p-value 13.1412% 0.1585%
beta-weight 0.8835

std error of regression 2.81544425
R-squared 78.05%
Adjusted R-squared 74.91%

Number of observations 9
Residual degrees of freedom 7

t-statistic for computing
95% confidence intervals 2.3646

a. write the regression equation
b. how much does the average profit from a project increase when the team spends one more hour on a project?
c. Using α = 0.10 can you reject the null hypothesis that the true value of the constant in the regression model is not zero? Explain.
d. Using α = 0.05 can you reject the null hypothesis that the true value of the coefficient of the Time variable is not zero? Explain
e. If the team will work 100 hours on a project what will be the expected or mean profit from the project?
f. Write a hypothesis to test the claim that each extra hour spent working on the project increases profitability of that project by less than $250.

Answer

a. write the regression equation

Profit = 3.35738712+1.8163041 *Time

b. How much does the average profit from a project increase when the team spends one more hour on a project?

This is given by the regression coefficient of time

β1 =1.8163041

c. Using α = 0.10 can you reject the null hypothesis that the true value of the constant in the regression model is not zero? Explain.

The critical value of for the null hypothesis (H0:β0 = 0 Vs H1: β0 ≠ 0) is
. The calculated value of t = 1.7079. The null hypothesis can be accepted as the calculated value is less than the critical value.

Please note the null hypothesis and alternatives specified in the answer. It is wrong to set null hypothesis as H0: β0 ≠ 0

d. Using α = 0.05 can you reject the null hypothesis that the true value of the coefficient of the Time variable is not zero? Explain

The critical value of for the null hypothesis (H0:β1 = 0 Vs H1: β1 ≠ 0) is
. The calculated value of t = 4.9888. The null hypothesis is rejected as the calculated value is greater than the critical value.

Please note the null hypothesis and alternatives specified in the answer. It is wrong to set null hypothesis as H0: β1 ≠ 0

e. If the team will work 100 hours on a project what will be the expected or mean profit from the project?

Mean profit when time = 100 hrs

Profit = 3.35738712+1.8163041 *100 = 184.9878

f. Write a hypothesis to test the claim that each extra hour spent working on the project increases profitability of that project by less than $250.

H0: Regression coefficient of time = 0.250 (H0:β1 = 0.250)
H1: Regression coefficient of time < 0.250 (H0:β1 < 0.250)

2. Regression: Profits
Constant Energy cost
Coefficient 300.156701 17.14956955
Std error of coef 290.462959 6.075477932
t-ratio 1.0334 2.8228
p-value 30.7795% 0.7458%
beta-weight 0.4119

std error of regression 392.6116924
R-squared 16.96%
Adjusted R-squared 14.84%

Number of observations 41
Residual degrees of freedom 39

t-statistic for computing
95% confidence intervals 2.0227

a. Given this output, what is an estimate for the change in price of a refrigerator model when its annual energy costs decrease by $20?
The regression equation is
Profit = 300.156701 + 17.1495695 *Energy cost
When Energy cost decrease by $ 20
Profit = 300.156701 + 17.1495695* ( -20) =$ - 42.834

b. Given this estimate, would you go ahead with the new technology? Explain.
The profit is negative (loss) when the energy cost decrease by $20. The management should consider other factors before implementing the project.

The model adequacy is given by R 2 value. Here R 2 = 0.1696. Thus model Explain only 16.96 % total variation in profit. Thus 83.04 % variation is profit is determined by other factors. Thus even though cost have an impact on profit, it have only a small role in explaining the variations in profit.

c. Does this estimate make sense? Explain.

Here we test the null hypothesis that regression coefficient of energy cost is zero.
H0:β1 = 0 Vs H1: β1 ≠ 0

The critical value at α = .05 is 2.023. The null hypothesis that β1 = 0 is rejected as the calculated value of t ( = 2.8228) is greater than the critical value. Thus Energy cost has a significant impact on profit.