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 = 0.232.

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

s² = 0.13.

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 for testing or estimating σ² or σChi-square goodness-of-fit    Chi-square test of independenceOne-way ANOVATwo-way ANOVAChi-square test of homogeneityF test for two variances

(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

Given,

A random sample of 26 roller bearings from the old manufacturing process showed the sample variance of diameters to be

s²= 0.232.

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

s² = 0.13.

F test for two variances;

(i) Give the value of the level of significance = 5%

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

Null hypothesis : there is no difference (either way) in the population variances between the old and new manufacturing processes;

H0: σ1² = σ2²

Alternate Hypothesis : there is a difference (either way) in the population variances between the old and new manufacturing processes

H1: σ1² σ2²

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

Given
n1 : Sample Size :Old Manfacturing process	26
n2 : Sample Size of new Manfacturing process	29
: Sample Varince of diameter:Old Manfacturing process	0.232
: Sample Varince of diameter:New Manfacturing process	0.13
Degrees of Freedom : Numerator : n1 -1;26-1=25	25
Degrees of Freedom : Denominator : n2 -1;29-1=28	28
Level of Significance :	5%

sample test statistic = 1.7846

(iii)

For Two tailed test :

0.100 < P-value < 0.200

(iv) Conclude the test.

p-value : 0.139 >  ​ : 0.05

Since the P-value is greater than or equal to the level of significance = 0.05, we fail to 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