Question

The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars.

Predictor	Coefficient			SE Coefficient			t			p-value
Constant		7.166			3.075			2.330			0.010
x1		0.228			0.302			0.755			0.000
x2	−	1.184			0.586		−	2.020			0.028
x3	−	0.193			0.110		−	1.755			0.114
x4		0.572			0.293			1.952			0.001
x5	−	0.056			0.022		−	2.545			0.112

Analysis of Variance
Source	DF		SS		MS		F		p-value
Regression	5		1,847.24		369.4		5.60		0.000
Residual Error	41		2,704.20		65.96
Total	46		4,551.44

x₁ is the number of architects employed by the company.

x₂ is the number of engineers employed by the company.

x₃ is the number of years involved with health care projects.

x₄ is the number of states in which the firm operates.

x₅ is the percent of the firm’s work that is health care−related.

1. Write out the regression equation. (Negative answers should be indicated by a minus sign. Round your answers to 3 decimal places.)

2. How large is the sample? How many independent variables are there?

1. c-1. At the 0.05 significance level, state the decision rule to test: H₀: β₁ = β₂ = β₃ =β₄ = β₅ = 0; H₁: At least one β is 0. (Round your answer to 2 decimal places.)

1. c-2. Compute the value of the F statistic. (Round your answer to 2 decimal places.)

1. c-3. What is the decision regarding H₀: β₁ = β₂ = β₃ = β₄ = β₅ = 0?

1. d-1. State the decision rule for each independent variable. Use the 0.05 significance level. (Round your answers to 3 decimal places.)

For x1		For x2		For x3		For x4		For x5
H0: β1 = 0		H0: β2 = 0		H0: β3 = 0		H0: β4 = 0		H0: β5 = 0
H1: β1 ≠ 0		H1: β2 ≠ 0		H1: β3 ≠ 0		H1: β4 ≠ 0		H1: β5 ≠ 0

1. d-2. Compute the value of the test statistic. (Negative answers should be indicated by a minus sign. Round your answers to 3 decimal places.)

1. d-3. For each variable, make a decision about the hypothesis that the coefficient is equal to zero.

Answer 1

We have given :

The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars.

x₁ is the number of architects employed by the company.

x₂ is the number of engineers employed by the company.

x₃ is the number of years involved with health care projects.

x₄ is the number of states in which the firm operates.

x₅ is the percent of the firm’s work that is health care−related.

## a ) Write out the regression equation. (Negative answers should be indicated by a minus sign. Round your answers to 3 decimal places.)

Answer

: ŷ = 7.166 + ( 0.228 * x1 ) - ( 1.184 * x2 ) - ( 0.193 * x3) + ( 0.572 * x4) - ( 0.056 *x5)

## b) How large is the sample? How many independent variables are there?

Answer : sample size is 47 and , there are 5 independent variables .

# c-1 )   At the 0.05 significance level, state the decision rule to test: H₀: β₁ = β₂ = β₃ =β₄ = β₅ = 0; H₁: At least one β is 0. (Round your answer to 2 decimal places.)

Answer :

Decision :

we reject Ho if p value less than α value using p value approach , here p value is  less than α value we reject Ho at given level of significance .

## c-2. Compute the value of the F statistic. (Round your answer to 2 decimal places.)

Answer : F statistics = 5. 60

## c-3. What is the decision regarding H₀: β₁ = β₂ = β₃ = β₄ = β₅ = 0?

Answer ;

Decision :

we reject Ho if p value less than α value using p value approach , here p value is  less than α value we reject Ho at given level of significance .

Conclusion :

There is sufficient evidence to conclude that at least one coefficient is differ , that is result is significant we can say overall regression model significant at given level of significance .

## d-1. State the decision rule for each independent variable. Use the 0.05 significance level. (Round your answers to 3 decimal places.)

Answer :

for each independent variable , we can use p value approach , reject Ho if p value is less than α value .

For x1 : Ho : β₁ = 0 VS H1 :β₁ ≠ 0

For x2 : Ho : β₂ = 0 VS H1 :β2 ≠ 0

For x3 : Ho : β₃ = 0 VS H1 :β3 ≠ 0

For x4 : Ho : β₄ = 0 VS H1 :β4 ≠ 0

For x5 : Ho : β₅ = 0 VS H1 :β5 ≠ 0

## d-2. Compute the value of the test statistic. (Negative answers should be indicated by a minus sign. Round your answers to 3 decimal places.)

Answer :

t test statistics for x1 = 0.755

t test statistics for x2 = - 2.020

t test statistics for x3 = - 1.755

t test statistics for x4 = 1.952

t test statistics for x5 = - 2.545

   

## d-3. For each variable, make a decision about the hypothesis that the coefficient is equal to zero.

Answer :

# for x1

Decision : here p value is 0.00 <  α , here we reject Ho

Conclusion :

There is sufficient evidence to conclude that coefficient of x1 is significant at given level of significance .

# for x2

Decision : here p value is ( 0.028)  <  α , here we reject Ho

Conclusion :

There is sufficient evidence to conclude that coefficient of x2 is significant at given level of significance .

# for x3

Decision : here p value is ( 0 .114 )  >   α , here we fail to reject Ho

Conclusion :

There is Insufficient evidence to conclude that coefficient of x3 is significant at given level of significance . ( we accept Ho , coefficient of x3 is not significant )

# for x4

Decision : here p value is ( 0 .001 ) < α , here we fail to reject Ho

Conclusion :

There is sufficient evidence to conclude that coefficient of x3 is significant at given level of significance .  

# for x5

Decision : here p value is ( 0 .112 )  >   α , here we fail to reject Ho

Conclusion :

There is Insufficient evidence to conclude that coefficient of x5 is significant at given level of significance . ( we accept Ho , coefficient of x5 is not significant )