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Shulman Steel Corporation makes bearings that are supplied to other companies. One of the machines makes bearings that are supposed to have a diamcter of 4 inches. The bearings that have a diameter of either more or less than 4 inches are considered defective and are discarded. When working properly, the machine does not produce more than \(7 \%\) of bearings that are defective. The quality control inspector selects a sample of 200 bearings each week and inspects them for the size of their diameters. Using the sample proportion, the quality control inspector tests the null hypothesis \(p \leq .07\) against the alternative hypothesis \(p \geq\) 07, where \(p\) is the proportion of bearings that are defective. He always uses a \(2 \%\) significance level. If the null hypothesis is rejected, the machine is stopped to make any necessary adjustments. One sample of 200 bearings taken recently contained 22 defective bearings. a. Using the \(2 \%\) significance level, will you conclude that the machine should be stopped to make necessary adjustments? b. Perform the test of part a using a \(1 \%\) significance level. Is your decision different from the one in part a?

Step by step solution

01

Calculate the standard error

The standard error (SE) for a proportion is calculated using the formula \(\sqrt{p(1-p)/n}\) where \(p\) is the proportion under the null hypothesis and \(n\) is the sample size. Here, \(p = 0.07\) and \(n = 200\), so the SE is \(\sqrt{0.07 * (1-0.07) / 200}\).

02

Determine the test statistic

The test statistic (Z) for a proportion is calculated using the formula \((\hat{p} - p) / SE\), where \(\hat{p}\) is the sample proportion and \(SE\) is the standard error calculated in Step 1. Here, \(\hat{p} = 22/200 = 0.11\), so the Z score is \((0.11 - 0.07) / SE\).

03

Compare the test statistic to the critical Z score

At a 2% significance level, the critical Z score (for a one-tailed test) is 2.05. If the calculated Z score from Step 2 is greater than 2.05, the null hypothesis is rejected.

04

Repeat the test at a 1% significance level

Repeat Steps 1-3, but use a critical Z score of 2.33 (for a one-tailed test at a 1% significance level).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Hypothesis Test

A proportion hypothesis test helps us determine if there is enough evidence to support a specific claim about a population proportion. In our case, Shulman Steel Corporation is checking if the defect rate of their bearings exceeds the threshold of 7%. For this type of test, we'll compare the sample proportion of defective bearings (\( \hat{p} = \frac{22}{200} = 0.11 \)) to the assumed population proportion in the null hypothesis (\( p = 0.07 \)).
Conducting this test involves setting up hypotheses. The null hypothesis (\( H_0 \)) claims that the proportion of defective bearings is less than or equal to 7% (\( p \leq 0.07 \)), and the alternative hypothesis (\( H_a \)) posits that the defect rate is greater than this proportion (\( p > 0.07 \)). The decision to reject or fail to reject the null hypothesis is based on the comparison of the Z score with the critical Z score determined by the level of significance.

Standard Error Calculation

The standard error (SE) measures the variability of the sample proportion and is crucial in hypothesis testing. It's calculated using the formula:\[SE = \sqrt{\frac{p(1-p)}{n}}\]
For Shulman Steel Corporation, with a null hypothesis proportion (\( p = 0.07 \)) and a sample size (\( n = 200 \)), the standard error is:\[SE = \sqrt{\frac{0.07 \times (1 - 0.07)}{200}}\]
This calculation provides us with an idea of how much sampling variability to expect. A smaller SE indicates that our sample proportion is a more precise estimate of the actual population proportion.
It's a critical step to accurately determine the test statistic in the subsequent phase of hypothesis testing.

Significance Level

The significance level (\( \alpha \)) sets the threshold for deciding when to reject the null hypothesis. In the airline's quality control example, they first use a 2% significance level (\( \alpha = 0.02 \)). This means there is a 2% risk of incorrectly rejecting the null hypothesis if it's true.
Choosing a significance level is all about trade-offs:

• Higher significance levels lead to more type I errors (false positives).
• Lower significance levels increase the chance of type II errors (failing to reject a false null hypothesis).

It's essential to balance these risks based on the context of the issue. In critical quality control situations, a lower level might be more appropriate to ensure any deviation is caught.

Critical Z Score

The critical Z score is a key benchmark in hypothesis testing. It helps determine whether the calculated test statistic is extreme enough to reject the null hypothesis. For Shulman Steel, with a 2% significance level, the critical Z score is 2.05.
In a one-tailed test like this, we look at only one end of the distribution, considering only if the defect rate is significantly higher. If the calculated Z score (from the sample data) exceeds 2.05, it suggests enough evidence to conclude an issue with the bearings exceeds 7%, leading to machine adjustments.
When the significance level is changed to 1%, the critical Z score rises to 2.33, reflecting a stricter threshold. Understanding these metrics ensures decisions are made with a clear statistical basis.