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 = 0.214.

Another random sample of 29 roller bearings from the new manufacturing process showed the sample variance of their diameters to be

s² = 0.115.

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.

One-way ANOVAChi-square test of independence    F test for two variancesChi-square goodness-of-fitTwo-way ANOVAChi-square for testing or estimating σ² or σChi-square test of homogeneity

(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 > 0.2000.100 < P-value < 0.200    0.050 < P-value < 0.1000.020 < P-value < 0.0500.002 < P-value < 0.020P-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

A random sample of 21 roller bearings from the old manufacturing process showed the sample variance of diameters to be s2 = 0.214.

Another random sample of 29 roller bearings from the new manufacturing process showed the sample variance of their diameters to be s2 = 0.115.

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

.This problem is testing the equality of two population variances.

So it is solved using F test for two variances.

Answer:-  F test for two variances

(i) Give the value of the level of significance. = 0.05

test the claim that there is a difference (either way) in the population variances between the old and new manufacturing processes.

It is a two-sided test.

The null & alternative hypothesis is :

H0: σ12 = σ22;

Vs

Ha: σ12  ≠ σ22

The decision about the null hypothesis

Since from the sample information we get that FL​=0.423<F=1.861<FU​=2.232, it is then concluded that the null hypothesis is not rejected.

df1 = n1-1 = 20 ; df2 = n2-1 = 28

P-value = 2* P( Fdf1,df2 > F)

= 2* P( F20,28 > 1.861)

P-value = 2* 0.0641

P-value = 0.1281

Answer:- 0.100 < P-value < 0.20

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

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