Question

After a catastrophic failure of your injection mold die, the production team has rebuilt the equipment and production is running smoothly again. However, management wants to be sure the quality after the repairs is the same as before. Taking it on faith that σ = 0.030 grams before the failure, you conduct a quick experiment on a batch produced after the failure, and you measure s = 0.036 grams from a random sample of 25 O-rings.

a) Based on this experiment, can you conclude the injection mold process is exhibiting the same variance before and after the repair? Show all of your work  

b) The management brings in another production batch, produced before the failure, and wants you to compare the variances of this batch with the batch from part (a). You collect 30 random O-rings from this batch and measure the sample standard deviation, s. Assuming a 2-sided alternative, what is the lowest value of s you could measure and still be able to conclude the two batches have identical variances? Show all of your work.

Answer 1

a)

To test the injection mold process is exhibiting the same variance before and after the repair, we use Chi-square test for the variance.

The chi-square hypothesis test is

Null Hypothesis

versus

Alternative Hypothesis

To test the stated hypothesis, the test statistic is

Where

n = sample size

s = sample standard deviation

= target standard deviation

Reject null hypothesis, if

or

From the data,

n = 25

s = 0.036

= 0.030

The test statistic is calculated

For 5% level of significance and 24 (= 25-1) degrees of freedom chi-square critical values are

Since the calculated chi-square (=34.56) > chi-square tabulated (=12.401) we do not reject the null hypothesis and infer that injection mold process is exhibiting the same variance before and after the repair.

or

Also, the calculated chi-square (=34.56) < chi-square tabulated (=39.364) we do not reject the null hypothesis and infer that injection mold process is exhibiting the same variance before and after the repair.

b)

For chi-square 2-sided alternative test for variance, we do not reject null hypothesis if

Hence under null hypothesis, the lowest value of s that could measure and still be able to conclude the two batches have identical variances is

Given,

n = sample size = 30

From part (a)

The lowest value of s that could measure and still be able to conclude the two batches have identical variances is 0.0216