Question

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

Production Volume (units)	Total Cost ($)
400	4,000
450	4,900
550	5,400
600	6,000
700	6,300
750	6,900

This data was used to develop an estimated regression equation, ŷ = 1,305.33 + 7.44x, 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.

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.)

What is your conclusion?

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

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

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

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

3.

DETAILS

Answer 1

Solution

Final answers are given below. Back-up Theory and Details of calculations follow at the end.

Part (a)

Hypotheses.

Null H₀: The estimated regression line is not significant.

Alternative H_A: The estimated regression line is significant. Answer 1

Part (b)

ANOVA table.

ANOVA	α	0.05
Source	SS	df	MS	F	p-value
Regression	5189400	1	5189400	86.8761	0.0007
Error	238933.3333	4	59733.3333
Total	5428333.3333	5

Answer 2

Part (c)

Test statistic.

F = 86.8761 Answer 3

Part (d)

p-value = 0.0007 Answer 4

Part (e)

Conclusion

Since p-value < significance level, the null hypothesis cannot accepted. Hence, we conclude:

The estimated regression line is significant, implying that the linear model is adequate to predict Total cost based on Production volume. Answer 5

Back-up Theory and Details of calculations

The linear regression model: Y = β₀ + β₁X + ε, where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y_hat = β_0hat + β_1hatX,

where β_1hat = Sxy/Sxx and β_0hat = Ybar – β_1hat.Xbar.

For ANOVA,

SST = Syy; SSR = β²_1cap x Sxx; SSE = SST – SSR

Df : Total: number of observations – 1, Regression: 1, Error: Total – Regression df

F = MSR/MSE

Fcrit = upper α% point of F_{1, n – 2}, where α = significance level and n = number of observations.

p-value = P(F_{1, n – 2} > F).

Decision:

If F > Fcrit or equivalently, p-value < α, H₀ is rejected.

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i Sxx = Σ(i = 1 to n)(x_i – Xbar)² Syy = Σ(i = 1 to n)(y_i – Ybar)²  Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)}

Calculations

n	6
Xbar	575.00
ybar	5583.33
Sxx	93750.00
Syy	5428333.33
Sxy	697500.00
β1cap	7.44
β0cap	1305.3333

DONE