Question

Two processes for manufacturing large roller bearings are under study. In both cases, the diameters (in...

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.

Homework Answers

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.

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