Question

The variance in a production process is an important measure of the quality of the process. A large variance often signals an opportunity for improvement in the process by finding ways to reduce the process variance.

Machine 1

2.95	3.45	3.50	3.75	3.48	3.26	3.33	3.20
3.16	3.20	3.22	3.38	3.90	3.36	3.25	3.28
3.20	3.22	2.98	3.45	3.70	3.34	3.18	3.35
3.12

Machine 2

3.22	3.30	3.34	3.28	3.29	3.25	3.30	3.27
3.37	3.34	3.35	3.19	3.35	3.05	3.36	3.28
3.30	3.28	3.30	3.20	3.16	3.33

Conduct a statistical test to determine whether there is a significant difference between the variances in the bag weights for two machines. Use a 0.05 level of significance. What is your conclusion?

(a) State the null and alternative hypotheses.

A. H₀: σ₁² = σ₂²
H_a: σ₁² ≠ σ₂²

B. H₀: σ₁² ≠ σ₂²
H_a: σ₁² = σ₂²

C. H₀: σ₁² ≤ σ₂²
H_a: σ₁² > σ₂²

D. H₀: σ₁² > σ₂²
H_a: σ₁² ≤ σ₂²

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

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

State your conclusion.

A. Reject H₀. We can conclude that the process variances are significantly different.

B. Reject H₀. We cannot conclude that the process variances are significantly different.     

C. Do not reject H₀. We can conclude that the process variances are significantly different.

D. Do not reject H₀. We cannot conclude that the process variances are significantly different.

Which machine, if either, provides the greater opportunity for quality improvements?

A. Machine 1 offers the best opportunity for process quality improvements.

B. Machine 2 offers the best opportunity for process quality improvements.    

C. Neither machine offers an opportunity for process quality improvements.

Answer 1

A. H₀: σ₁² = σ₂²
H_a: σ₁² ≠ σ₂²

b)

n1    =	25
n2    =	22
numerator df dN =n1-1=		24
denominator df dD=n2-1 =		21
s12 =	0.049
s22 =	0.006

Test statistic =s12/s22 =

8.42

c)

P value =

0.0000

from excel: 2*fdist(8.42 ,24,21)

since p value <0.05

A. Reject H₀. We can conclude that the process variances are significantly different.

A. Machine 1 offers the best opportunity for process quality improvements. (since it has higher standard deviaiton)