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	3.28
3.38	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?

State the null and alternative hypotheses.

H₀: σ₁² ≠ σ₂²

H_a: σ₁² = σ₂²

H₀: σ₁² > σ₂²

H_a: σ₁² ≤ σ₂²

    

H₀: σ₁² = σ₂²

H_a: σ₁² ≠ σ₂²

H₀: σ₁² ≤ σ₂²

H_a: σ₁² > σ₂²

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

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

p-value =

State your conclusion.

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

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

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

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?

Machine 1 offers the best opportunity for process quality improvements.

Machine 2 offers the best opportunity for process quality improvements.    

Neither machine offers an opportunity for process quality improvements.

Answer 1

State the null and alternative hypotheses.

H₀: σ₁² = σ₂²

H_a: σ₁² ≠ σ₂²

Find the value of the test statistic.

8.29

p-value = 0.0000

State your conclusion.

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

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

Machine 2 offers the best opportunity for process quality improvements.    

The output is:

σ₁: standard deviation of Machine 1

σ₂: standard deviation of Machine 2

Ratio: σ₁/σ₂

F method was used. This method is accurate for normal data only.

Descriptive Statistics

Variable	N	StDev	Variance	95% CI for σ²
Machine 1	25	0.221	0.049	(0.030, 0.095)
Machine 2	22	0.077	0.006	(0.003, 0.012)

Ratio of Variances

Estimated Ratio	95% CI for Ratio using F
8.28901	(3.501, 19.155)

Test

Null hypothesis	H₀: σ₁² / σ₂² = 1
Alternative hypothesis	H₁: σ₁² / σ₂² ≠ 1
Significance level	α = 0.05

Method	Test Statistic	DF1	DF2	P-Value
F	8.29	24	21	0.000