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 21 roller bearings from the old manufacturing process showed the sample variance of diameters to be s2 = 0.224. Another random sample of 26 roller bearings from the new manufacturing process showed the sample variance of their diameters to be s2 = 0.121. 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. Chi-square test of independence Chi-square test of homogeneity Chi-square for testing or estimating σ2 or σ Two-way ANOVA Chi-square goodness-of-fit F test for two variances One-way ANOVA (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

F test for two variances

i) level of significance =0.05

H0: σ12 = σ22; H1: σ12 ≠ σ22

ii) sample test statistic =(0.224/0.121)=1.85

iii)P-value of the sample test statistic 0.100 < 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.