Question

Two processes for manufacturing large roller bearings are under study. In both cases, the diameters (in centimeters) are being examined. A random sample of 26 roller bearings from the old manufacturing process showed the sample variance of diameters to be s² = 0.231. Another random sample of 28 roller bearings from the new manufacturing process showed the sample variance of their diameters to be s² = 0.146. Use a 5% level of significance to test the claim that there is a difference (either way) in the population variances between the old and new manufacturing processes.

Classify the problem as being a Chi-square test of independence or homogeneity, Chi-square goodness-of-fit, Chi-square for testing or estimating σ² or σ, F test for two variances, One-way ANOVA, or Two-way ANOVA, then perform the following.

Two-way ANOVAF test for two variances    

Chi-square test of homogeneity

Chi-square for testing or estimating σ² or σ

Chi-square goodness-of-fit

One-way ANOVA

Chi-square test of independence

(i) Give the value of the level of significance.


State the null and alternate hypotheses.

H₀: σ₁² = σ₂²; H₁: σ₁² > σ₂²

H₀: σ₁² = σ₂²; H₁: σ₁² < σ₂²     

H₀: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

H₀: σ₁² < σ₂²; H₁: σ₁² = σ₂²

(ii) Find the sample test statistic. (Round your answer to two decimal places.)

(iii) Find the P-value of the sample test statistic.

P-value > 0.200

0.100 < P-value < 0.200    

0.050 < P-value < 0.100

0.020 < P-value < 0.050

0.002 < P-value < 0.020

P-value < 0.002

(iv) Conclude the test.

Since the P-value is greater than or equal to the level of significance α = 0.05, we fail to reject the null hypothesis.

Since the P-value is less than the level of significance α = 0.05, we reject the null hypothesis.    

Since the P-value is less than the level of significance α = 0.05, we fail to reject the null hypothesis.

Since the P-value is greater than or equal to the level of significance α = 0.05, we reject the null hypothesis.

(v) Interpret the conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to show that the variance for the new manufacturing process is different.

At the 5% level of significance, there is sufficient evidence to show that the variance for the new manufacturing process is not different.    

At the 5% level of significance, there is insufficient evidence to show that the variance for the new manufacturing process is not different.

At the 5% level of significance, there is sufficient evidence to show that the variance for the new manufacturing process is different.

Answer 1

The statistical software output for this problem is:

Hence,

Two-way ANOVA F test for two variances    

i) Level of significance = 0.05

Hypotheses: H₀: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

ii) Test statistic = 1.58

iii) P-value > 0.200

iv) Since the P-value is greater than or equal to the level of significance α = 0.05, we fail to reject the null hypothesis.

v) At the 5% level of significance, there is insufficient evidence to show that the variance for the new manufacturing process is different.