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Chapter 9: Problem 5

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is \(<.05 \mathrm{~mm}\). What hypotheses should be tested, and why? In this context, what are the type I and type II errors?

Short Answer

Expert verified
Test \( H_0: \sigma \geq 0.05 \text{ mm} \) against \( H_a: \sigma < 0.05 \text{ mm} \). Type I error: reject \( H_0 \) when \( \sigma \geq 0.05 \). Type II error: fail to reject \( H_0 \) when \( \sigma < 0.05 \).

Step by step solution

01

Define the Null and Alternative Hypotheses

To determine if the standard deviation of sheath thickness is less than 0.05 mm, we set up the hypotheses related to the standard deviation, \( \sigma \). The null hypothesis \( H_0 \) represents the current belief or default state, while the alternative hypothesis \( H_a \) represents what we want to prove.\( H_0: \sigma \geq 0.05 \text{ mm} \) and \( H_a: \sigma < 0.05 \text{ mm} \). The alternative hypothesis is what the company wants to prove.
02

Explain Type I and Type II Errors

In hypothesis testing, errors can occur in our decision process. A Type I error happens when we reject the null hypothesis \( H_0 \) when it is actually true, meaning we incorrectly conclude that the standard deviation is less than 0.05 mm when it is not. A Type II error occurs when we fail to reject the null hypothesis when the alternative hypothesis is true, meaning we incorrectly conclude that the standard deviation is not less than 0.05 mm when it actually is.

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Key Concepts

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

Null Hypothesis
In the context of hypothesis testing, the null hypothesis is essentially a statement of no effect or no difference. It represents the default or existing belief about a population parameter.
In our given exercise, the null hypothesis, denoted as \( H_0 \), is that the true standard deviation of the polyethylene sheaths' thickness is greater than or equal to 0.05 mm.
This hypothesis functions as a kind of baseline that we work against. By testing this hypothesis, the company checks if their existing understanding (that the standard deviation might be 0.05 mm or more) holds true.
  • The null hypothesis is sometimes thought of as the "status quo."
  • You only reject it if you have sufficient statistical evidence against it.
  • If the evidence is lacking, you continue to accept the null hypothesis as true.
Alternative Hypothesis
The alternative hypothesis is the statement you want to be able to conclude if the data provides sufficient evidence against the null hypothesis. In our example, the alternative hypothesis is denoted as \( H_a \).
For our exercise, \( H_a \) states that the true standard deviation of sheath thickness is less than 0.05 mm.
This hypothesis aligns with what the company wants to confirm: that the standard deviation meets their quality standards and is smaller than their specified requirement.
  • Think of it as the hypothesis you want to support with evidence.
  • If the null hypothesis is rejected, the alternative hypothesis is accepted instead.
  • This principle helps to guide decisions in scientific research or quality control tasks.
Type I Error
In the hypothesis testing process, errors can occur due to incorrect conclusions. A Type I error happens when you reject the null hypothesis \( H_0 \) when it is actually true.
For this exercise, making a Type I error means that the company concludes that the standard deviation is less than 0.05 mm when it is not.
This error leads to a false positive result.
  • A Type I error can lead to unnecessary changes or actions based on incorrect conclusions.
  • It's also referred to as "false alarm" or "alpha error."
  • Controlling the rate of Type I errors is crucial and is usually done by setting a significance level, typically 0.05 or 5%.
Type II Error
A Type II error occurs in hypothesis testing when you fail to reject the null hypothesis \( H_0 \) when the alternative hypothesis \( H_a \) is true.
In our exercise, committing a Type II error means the company concludes that the standard deviation is not less than 0.05 mm, even though it actually is.
This results in a false negative outcome.
  • Type II errors are also known as "beta errors."
  • They often lead to missed opportunities for improvement or necessary change.
  • The probability of a Type II error is denoted by \( \beta \), and reducing \( \beta \) usually necessitates increasing the sample size.

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Most popular questions from this chapter

For the following pairs of assertions, indicate which do not comply with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples): a. \(H_{0}: \mu=100, H_{\mathrm{a}}: \mu>100\) b. \(H_{0}: \sigma=20, H_{\mathrm{a}}: \sigma \leq 20\) c. \(H_{0}: p \neq .25, H_{\mathrm{a}}: p=.25\) d. \(H_{0}: \mu_{1}-\mu_{2}=25, H_{\mathrm{a}}: \mu_{1}-\mu_{2}>100\) e. \(H_{0}: S_{1}^{2}=S_{2}^{2}, H_{a}: S_{1}^{2} \neq S_{2}^{2}\) f. \(H_{0}: \mu=120, H_{\mathrm{a}}: \mu=150\) g. \(H_{0}: \sigma_{1} / \sigma_{2}=1, H_{\mathrm{a}}: \sigma_{1} / \sigma_{2} \neq 1\) h. \(H_{0}: p_{1}-p_{2}=-.1, H_{\mathrm{a}}: p_{1}-p_{2}<-.1\)

The amount of shaft wear (.0001 in.) after a fixed mileage was determined for each of \(n=8\) internal combustion engines having copper lead as a bearing material, resulting in \(\bar{x}=3.72\) and \(s=1.25\). a. Assuming that the distribution of shaft wear is normal with mean \(\mu\), use the \(t\) test at level . 05 to test \(H_{0}: \mu=3.50\) versus \(H_{\mathrm{a}}: \mu>3.50\). b. Using \(\sigma=1.25\), what is the type II error probability \(\beta\left(\mu^{\prime}\right)\) of the test for the alternative \(\mu^{\prime}=4.00\) ?

An aspirin manufacturer fills bottles by weight rather than by count. Since each bottle should contain 100 tablets, the average weight per tablet should be 5 grains. Each of 100 tablets taken from a very large lot is weighed, resulting in a sample average weight per tablet of \(4.87\) grains and a sample standard deviation of \(.35\) grain. Does this information provide strong evidence for concluding that the company is not filling its bottles as advertised? Test the appropriate hypotheses using \(\alpha=.01\) by first computing the \(P\)-value and then comparing it to the specified significance level.

A spectrophotometer used for measuring \(\mathrm{CO}\) concentration [ppm (parts per million) by volume] is checked for accuracy by taking readings on a manufactured gas (called span gas) in which the \(\mathrm{CO}\) concentration is very precisely controlled at \(70 \mathrm{ppm}\). If the readings suggest that the spectrophotometer is not working properly, it will have to be recalibrated. Assume that if it is properly calibrated, measured concentration for span gas samples is normally distributed. On the basis of the six readings- \(85,77,82,68,72\), and 69-is recalibration necessary? Carry out a test of the relevant hypotheses using the \(P\)-value approach with \(\alpha=.05\).

Pairs of \(P\)-values and significance levels, \(\alpha\), are given. For each pair, state whether the observed \(P\) value would lead to rejection of \(H_{0}\) at the given significance level. a. \(P\)-value \(=.084, \alpha=.05\) b. \(P\)-value \(=.003, \alpha=.001\) c. \(P\)-value \(=.498, \alpha=.05\) d. \(P\)-value \(=.084, \alpha=.10\) e. \(P\)-value \(=.039, \alpha=.01\) f. \(P\)-value \(=.218, \alpha=.10\)
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