You work for Northwest Fasteners, Inc., a manufacturing company that makes rivet

Question

You work for Northwest Fasteners, Inc., a manufacturing company that makes rivets for construction ofcommercial aircraft. The rivets are made from an aluminum alloy and have very strict strength requirements. Your contract with Boeing states that 97% of the rivets must pass a strength test. It does not cost very much to test a rivet. You just put it into a machine and the machine stresses it until it breaks, then the machine shows the breaking strength of the rivet. Your manager wants you to provide evidence that your rivets meet the 97% contract requirements so you think about testing all the rivets that come off the line. Fortunately, you noticed that the test for the rivets is destructive, meaning the rivet is destroyed in the test. That means you can't really test them all.

Your group is tasked with coming up with a plan to provide evidence that your production process meets the strength requirements 97% of the time.

Write a proposal (a general outline) to your manager of your plan and explain how it would provide evidence that you are meeting the contract requirements.

Explanation / Answer

One sample z-test

You work for Northwest Fasteners, Inc., a manufacturing company that makes rivets for construction of commercial aircraft. The rivets are made from an aluminium alloy and have very strict strength requirements. Your contract with Boeing states that 97% of the rivets must pass a strength test. It does not cost very much to test a rivet. You just put it into a machine and the machine stresses it until it breaks, then the machine shows the breaking strength of the rivet. Your manager wants you to provide evidence that your rivets meet the 97% contract requirements so you think about testing all the rivets that come off the line. Fortunately, you noticed that the test for the rivets is destructive, meaning the rivet is destroyed in the test. That means you can't really test them all.

Your group is tasked with coming up with a plan to provide evidence that your production process meets the strength requirements 97% of the time.

Write a proposal (a general outline) to your manager of your plan and explain how it would provide evidence that you are meeting the contract requirements.

Solution:

It is given that the contract with Boeing states that 97% of the rivets must pass a strength test. Here, we have to check the claim of the company whether the breaking strength of the rivet is 97% or not. For checking this claim or hypothesis we have to use the one sample z test for the population proportion. The null and alternative hypothesis for this test is given as below:

Null hypothesis: H0: The proportion of the breaking strength is 0.97 for the contract requirements.

Alternative hypothesis: Ha: The proportion of the breaking strength is less than 0.97 for the contract requirements.

Symbolically, we can write this hypothesis as below:

H0: p = 0.97 versus Ha: p < 0.97

This is a one tailed or lower tailed test.

We will assume the level of significance or the value of the alpha as 0.05 or 5%.

We will test some samples with the size more than 30 and we will find out the sample proportion for the breaking strength of the rivets.

After finding out the sample proportion, we will use the one sample z test for the proportion and then by using the t test statistic formula for the one sample z test for the proportion, we will find the p-value for the given test. Also, based on the given level of significance or alpha value, we will find the critical value for the given test.

For the 5% level of significance, the critical value is given as z = 1.645

The test statistic formula for the one sample z test is given as below:

Z = (sample proportion – 0.97) / sqrt (n*0.97*0.03)

Where n is the sample size.

After finding the test statistic value z, we can find the p-value by using the z-table.

Then we can take decision based on the p-value by using the following decision rule.

We reject the null hypothesis if the p-value is less than the given level of significance or the alpha value and we do not reject the null hypothesis if the p-value is greater than the given level of significance or the alpha value.

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