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TV safety The manufacturer of a metal stand for home TV sets must be sure that its product will not fail under the weight of the TV. since some larger scis weigh nearly 300 pounds, the company's safety inspectors have set a standard of ensuring that the stands can support an average of over 500 pounds. Their inspectors regularly subject a random sample of the stands to increasing weight until they fail. They test the hypothesis \(\mathrm{H}_{0}: \mu=500\) against \(\mathrm{H}_{\mathrm{A}}: \mu>500,\) using the level of significance \(\alpha=0.01 .\) If the sample of stands fails to pass this safety test, the inspectors will not certify the product for sale to the general public. a) Is this an upper-tail or lower-tail test? In the context of the problem, why do you think this is important? b) Explain what will happen if the inspectors commit a Type I error. c) Explain what will happen if the inspectors commit a Type II error.

Step by step solution

01

Identify the Hypothesis Test Type

This hypothesis test is to determine whether the average weight the stands can support is greater than 500 pounds. We are testing \(H_0: \mu = 500\) against \(H_A: \mu > 500\). This setup defines an upper-tail test because we are interested in proving the mean is greater than a certain value.

02

Understanding Test Significance

The significance level \(\alpha = 0.01\) indicates we are using a 1% criterion for rejecting the null hypothesis. This means if our test statistic falls in the upper 1% tail of the distribution, we reject \(H_0\). It is important because it reflects the company's conservative approach to certifying stands safe under heavier TVs, minimizing the chance of false positives.

03

Explaining Type I Error

A Type I error occurs when we reject the null hypothesis \(H_0\) when it is true. In this context, it would mean concluding that the stands can support more than 500 pounds when they actually cannot. This could lead to product failures when used for heavier TVs, causing safety hazards.

04

Explaining Type II Error

A Type II error occurs when we fail to reject \(H_0\) when the alternative hypothesis \(H_A\) is true. In this situation, it would mean concluding that the stands cannot safely support more than 500 pounds when they actually can. While this would keep defective stands off the market, it may unnecessarily halt the sale of a reliable product.

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Key Concepts

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

Upper-tail test

When performing a hypothesis test, determining if it's an upper-tail or lower-tail test is crucial, as it defines the direction of the test. In the case of the TV stand manufacturer, they are conducting an upper-tail test. This is because the company aims to ensure that their stands can support more than the minimum safety standard of 500 pounds. Therefore, they are interested in finding evidence that the mean weight capacity, denoted as \( \mu \), is actually greater than 500 pounds.
Understanding this scenario helps us evaluate how far the sample mean has gone beyond the hypothesized mean of 500 pounds. Such a test is critical for the safety of the product because proving the stand capabilities exceeds this threshold can avert failures that might occur with heavier TVs.

Type I error

A Type I error happens when the null hypothesis, which is actually true, is incorrectly rejected. In simpler terms, it's a false alarm — believing there is an effect or difference when there isn't one. For the TV stand manufacturer, committing a Type I error would mean they mistakenly conclude that the TV stands can hold more than 500 pounds, when in reality, they do not. This could lead to severe consequences:

• The stands might fail when used with a heavy TV, posing safety hazards and potential injury risks.
• Faulty certification of the product may occur, leading to widespread failures and reputational damage for the company.
• Ultimately, it might complicate product liability issues.

Type II error

In contrast to a Type I error, a Type II error occurs when the null hypothesis is not rejected even though the alternative hypothesis is true. Essentially, it's like missing an alarm — overlooking a real effect or difference. In the context of the TV stands, a Type II error would mean the inspectors conclude that the stands cannot support more than 500 pounds when, in fact, they can.
While this error is less hazardous than a Type I, it still has its downsides:

• The manufacturer might unnecessarily restrict a product that could safely be certified, leading to lost sales potential.
• It could delay product availability in the market, affecting business operations and profitability.
• 91Ó°ÊÓ and time spent in re-evaluating testing processes can be significant.

Significance level

The significance level, denoted as \( \alpha \), is an important component in hypothesis testing. It's the threshold at which the null hypothesis is rejected. For the TV stand manufacturer, they have set \( \alpha = 0.01 \), meaning they are applying a stringent 1% criterion for the decision-making process.
This indicates that there is only a 1% chance of making a Type I error — wrongly concluding that their stands can support more than 500 pounds when they can't. Such a low significance level reflects the company's focus on safety and the high stakes involved:

• Ensuring that the stands are truly capable of holding heavier weights minimizes safety hazards.
• It reflects a conservative approach, prioritizing consumer safety and product reliability over unnecessary risk-taking.
• Aligning business reputation with reliability may also enhance consumer trust and brand loyalty.