In: Statistics and Probability
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 = 0.234.
Another random sample of 27 roller bearings from the new manufacturing process showed the sample variance of their diameters to be
s2 = 0.125.
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 ANOVA F test for two variances Chi-square for testing or estimating σ2 or σ One-way ANOVA Chi-square test of homogeneity Chi-square goodness-of-fit 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 > σ22H0: σ12 = σ22; H1: σ12 ≠ σ22 H0: σ12 < σ22; H1: σ12 = σ22H0: 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.