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Catheters During an angiogram, heart problems can be examined via a small tube (a catheter) threaded into the heart from a vein in the patient's leg. It's important that the company that manufactures the catheter maintain a diameter of \(2.00 \mathrm{mm}\). (The standard deviation is quite small.) Each day, quality control personnel make several measurements to test \(\mathrm{H}_{0}: \mu=2.00\) against \(\mathrm{H}_{\mathrm{A}}: \mu \neq 2.00\) at a significance level of \(\alpha=0.05 .\) If they discover a problem, they will stop the manufacturing process until it is corrected. a) Is this a one-sided or two-sided test? In the context of the problem, why do you think this is important? b) Explain in this context what happens if the quality control people commit a Type I error. c) Explain in this context what happens if the quality control people commit a Type II error.

Step by step solution

01

Understanding the Type of Test

The problem statement presents two hypotheses: the null hypothesis, \(H_0: \mu = 2.00\), and the alternative hypothesis, \(H_A: \mu eq 2.00\). The alternative hypothesis suggests deviations in either direction from the mean of 2.00 mm, indicating a two-sided test. This is important because any significant deviation, whether higher or lower than 2.00 mm, could affect the catheter's functionality and safety.

02

Analyzing Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected. In this context, it means the quality control personnel conclude there is a problem with the manufacturing process (diameter not equal to 2.00 mm) when, in fact, the true mean diameter is 2.00 mm. This could lead to unnecessary halting of production and economic loss.

03

Analyzing Type II Error

A Type II error happens when the null hypothesis is not rejected even though the alternative hypothesis is true. Here, it means failing to detect a manufacturing issue when the average catheter diameter is not 2.00 mm. This might result in defective catheters being produced and delivered, potentially risking patients' safety during angiograms.

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Key Concepts

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

Type I Error

In the context of hypothesis testing, a Type I error is often referred to as a 'false positive.' This occurs when the null hypothesis is incorrectly rejected, even though it is true. In the exercise about catheters, the null hypothesis is that the average diameter of the catheter is exactly 2.00 mm. When a Type I error occurs, the quality control team wrongly concludes that the average diameter is different from 2.00 mm.
This kind of error has significant implications:

• The production process may be halted unnecessarily. This can lead to unwarranted delays and economic losses.
• 91Ó°ÊÓ are wasted on investigating a non-existent problem.
• There could be missed opportunities to produce and distribute functioning catheters, affecting the company's market position.

Thus, minimizing the risk of a Type I error is crucial for maintaining operation efficiency and reducing unnecessary costs.

Type II Error

Type II error in hypothesis testing is described as a 'false negative.' This occurs when the null hypothesis fails to be rejected even though it is false. In our context, it means the quality control personnel accept that the catheter diameter is 2.00 mm, when it actually is not.
This error is potentially dangerous due to the following reasons:

• Defective catheters might be undetected and shipped, posing potential health risks during medical procedures like angiograms.
• The company faces the risk of reputational damage if substandard products are released to the market.
• Long-term financial implications due to the cost of recalls and loss of customer trust.

Preventing Type II errors is vital to ensuring product safety and maintaining company integrity.

Two-sided Test

A two-sided test is a statistical method used when the interested party wants to detect deviations in both directions from a specified parameter value. In the catheter quality control exercise, the company sets the null hypothesis with a mean diameter of 2.00 mm, and the alternative hypothesis suggests that the diameter is not equal to 2.00 mm.
This approach is significant for several reasons:

• It accounts for any discrepancies, whether the average diameter is smaller or larger than 2.00 mm.
• Ensures comprehensive quality control to detect all forms of inconsistency that could compromise catheter effectiveness and safety.
• Protects against potential oversights that might occur if only one direction of deviation was considered.

By adopting a two-sided test, the company ensures they are vigilant about both increases and decreases in diameter, an essential factor in maintaining optimal product quality.