Question

You may need to use the appropriate technology to answer this question.

Consider the following sample of production volumes and total cost data for a manufacturing operation.

Production Volume (units)	Total Cost ($)
400	4,100
450	5,000
550	5,500
600	5,900
700	6,500
750	7,000

This data was used to develop an estimated regression equation,

ŷ = 1,342.67 + 7.52x,

relating production volume and cost for a particular manufacturing operation. Use

α = 0.05

to test whether the production volume is significantly related to the total cost. (Use the F test.)

State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₁ ≥ 0
H_a: β₁ < 0    H₀: β₀ ≠ 0
H_a: β₀ = 0H₀: β₀ = 0
H_a: β₀ ≠ 0H₀: β₁ = 0
H_a: β₁ ≠ 0

Set up the ANOVA table. (Round your p-value to three decimal places and all other values to two decimal places.)

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F	p-value
Regression
Error
Total

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

What is your conclusion?

Do not reject H₀. We cannot conclude that the relationship between production volume and total cost is significant.

Reject H₀. We cannot conclude that the relationship between production volume and total cost is significant.     

Reject H₀. We conclude that the relationship between production volume and total cost is significant.

Do not reject H₀. We conclude that the relationship between production volume and total cost is significant.

Answer 1

Solution) :

Here,

Production Volume is independent variable(X)

Total cost is dependent variable(Y) .

Enter data in SPSS software as given below:

In variable view:

Production Volume   as (scale) measure.

Total cost   as (scale) measure.

In variable view:

Enter the data for both columns

To get the entered data in output :

Steps= analyse --- reports ---case summaries ---select both variable in variables column---ok

	Production_Volume	Total_Cost
1	400	4100
2	450	5000
3	550	5500
4	600	5900
5	700	6500
6	750	7000

To perform simple regression in SPSS :

Steps : Analyse ---Regression---Linear---Dependent(Total cost)---Independent(Production volume)---method(enter)---Ok

Model Summary
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.986a	.972	.965	194.765
a. Predictors: (Constant), Production_Volume

R-square =0.972

Which means there is approximately 97.2% variability in Dependent variable(Total Cost) is explained by Independent Variable(Production Volume).

Coefficientsa
Model		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
		B	Std. Error	Beta
1	(Constant)	1342.667	374.300		3.587	.023
	Production_Volume	7.520	.636	.986	11.822	.000
a. Dependent Variable: Total_Cost

Here, from the above coefficient table ,we have

Intercept (β₀) = 1342.67

Slope (β₁)       = 7.52

The Regression Equation is :

Ŷ = β₀+β₁X

    =1342.67+7.52*X

Now,State the null and alternative hypotheses.

H₀: β₁ = 0   there is not significant relationship between production volume and total cost.
H_a: β₁ ≠ 0    there is significant relationship between production volume and total cost.

ANOVAa
Model		Sum of Squares	df	Mean Square	F	Sig.
1	Regression	5301600.00	1	5301600.00	139.76	.000b
	Residual	151733.33	4	37933.33
	Total	5453333.33	5
a. Dependent Variable: Total_Cost
b. Predictors: (Constant), Production_Volume

Test statistic Value ,F=139.76

p-value =0.000<0.05 ,which means that regression is significant at 5% level of significance.

Conclusion: since p value is less than 0.05

.     Reject H₀. We conclude that the relationship between production volume and total cost is significant.