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Chapter 8: Problem 9

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ}\), 50 water samples will be taken at randomly selected times and the temperature of each sample recorded. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ}\) versus \(H_{a}: \mu>150^{\circ}\). In the context of this situation, describe type I and type II errors. Which type of error would you consider more serious? Explain.

Short Answer

Expert verified
A Type II error is more serious, as it risks harming the ecosystem.

Step by step solution

01

Understand the Hypotheses

In this context, we have a null hypothesis \( H_0 : \mu = 150^\circ F \), which claims that the mean temperature of the discharged water is \( 150^\circ F \). The alternative hypothesis \( H_a : \mu > 150^\circ F \) suggests that the mean temperature is greater than \( 150^\circ F \). Testing these hypotheses will help determine if the plant adheres to regulations.
02

Define Type I Error

A Type I error occurs when the null hypothesis is true, but we incorrectly reject it. In this case, a Type I error would mean concluding that the mean water temperature exceeds \( 150^\circ F \) when in fact it is \( 150^\circ F \). This could lead to unnecessary regulatory actions despite being in compliance.
03

Define Type II Error

A Type II error happens when the null hypothesis is false, but we fail to reject it. Here, a Type II error would mean failing to detect that the mean water temperature is actually greater than \( 150^\circ F \), hence incorrectly concluding that the plant is in compliance.
04

Evaluate Error Severity

In this context, a Type II error would likely be more serious. If the mean temperature truly exceeds \( 150^\circ F \) and we fail to discover this, the consequence could be significant harm to the river's ecosystem, whereas a Type I error would mostly result in a false alarm and potentially unnecessary operational adjustments.

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Key Concepts

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

Null Hypothesis
In the world of statistics, the null hypothesis, denoted as \( H_0 \), is a critical concept we use to test assumptions. Often, the null hypothesis states that there is no effect or no difference, serving as a starting point for our hypothesis testing. In our water temperature scenario, the null hypothesis \( H_0 : \mu = 150^{\circ} F \) posits that the mean temperature of the discharged water is precisely \( 150^{\circ} F \). This hypothesis represents the state of compliance according to environmental regulations. Before any testing begins, the assumption is that the power plant is not exceeding thermal limits.

When performing hypothesis testing, your goal is to determine whether to accept the null hypothesis or reject it in favor of an alternative hypothesis. Rejecting \( H_0 \) suggests that there is sufficient evidence to conclude an effect or a difference that wasn't initially assumed, such as the mean temperature being above \( 150^{\circ} F \).
Type I Error
A Type I error is a false positive conclusion in hypothesis testing, where we mistakenly reject a true null hypothesis. In simpler terms, we see an effect that doesn't exist. In our context, making a Type I error means concluding that the river's discharge temperature is over \( 150^{\circ} F \) when it is actually compliant with this limit. This could result in unnecessary regulatory penalties or operational changes for the plant.

This type of error relates directly to the significance level, often denoted by \( \alpha \), in hypothesis testing. The significance level is the probability threshold for committing a Type I error, commonly set at \( 0.05 \) or \( 5\% \). It's imperative to choose \( \alpha \) wisely, balancing the desire to avoid false positives with the potential implications of changes that may not be needed.
Type II Error
In contrast to a Type I error, a Type II error, represented by \( \beta \), happens when we fail to reject a false null hypothesis. Here, we are missing an effect that is actually present. Considering our scenario, a Type II error would mean not recognizing that the mean water temperature exceeds the regulatory \( 150^{\circ} F \), believing the plant is compliant when it's not. This oversight can lead to ecological consequences for the river system, such as harm to aquatic life and ecosystem balance.

The risk of committing a Type II error is related to the test's power, which is the probability of correctly rejecting a false null hypothesis. Increasing the sample size or using more precise measurements can help reduce \( \beta \), enhancing your confidence in detecting real effects, such as heightened water temperatures.

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Most popular questions from this chapter

A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than \(1300 \mathrm{KN} / \mathrm{m}^{2}\). The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for specimens of this mixture is normally distributed with \(\sigma=60\). Let \(\mu\) denote the true average compressive strength. a. What are the appropriate null and alternative hypotheses? b. Let \(\bar{X}\) denote the sample average compressive strength for \(n=10\) randomly selected specimens. Consider the test procedure with test statistic \(\bar{X}\) itself (not standardized). If \(\bar{x}=1340\), should \(H_{0}\) be rejected using a significance level of \(.01\) ? [Hint: What is the probability distribution of the test statistic when \(H_{0}\) is true?] c. What is the probability distribution of the test statistic when \(\mu=1350\) ? For a test with \(\alpha=.01\), what is the probability that the mixture will be judged unsatisfac- tory when in fact \(\mu=1350\) (a type II error)?

The calibration of a scale is to be checked by weighing a \(10-\mathrm{kg}\) test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with \(\sigma=.200 \mathrm{~kg}\). Let \(\mu\) denote the true average weight reading on the scale. a. What hypotheses should be tested? b. With the sample mean itself as the test statistic, what is the \(P\)-value when \(\bar{x}=9.85\), and what would you conclude at significance level .01? c. For a test with \(\alpha=.01\), what is the probability that recalibration is judged unnecessary when in fact \(\mu=\) 10.1? When \(\mu=9.8\) ?

The accompanying data on cube compressive strength (MPa) of concrete specimens appeared in the article "Experimental Study of Recycled Rubber-Filled High- Strength Concrete" (Magazine of Concrete \(\operatorname{Res}_{*}\) 2009: 549-556): \(\begin{array}{rrrrr}112.3 & 97.0 & 92.7 & 86.0 & 102.0 \\ 99.2 & 95.8 & 103.5 & 89.0 & 86.7\end{array}\) a. Is it plausible that the compressive strength for this type of concrete is normally distributed? b. Suppose the concrete will be used for a particular application unless there is strong evidence that true average strength is less than 100 MPa. Should the concrete be used? Carry out a test of appropriate hypotheses.

A random sample of 150 recent donations at a certain blood bank reveals that 82 were type A blood. Does this suggest that the actual percentage of type A donations differs from \(40 \%\), the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of .01. Would your conclusion have been different if a significance level of 05 had heen nsed?

Body armor provides critical protection for law enforcement personnel, but it does affect balance and mobility. The article "'Impact of Police Body Armour and Equipment on Mobility" (Applied Ergonomics, 2013: 957-961) reported that for a sample of 52 male enforcement officers who underwent an acceleration task that simulated exiting a vehicle while wearing armor, the sample mean was \(1.95 \mathrm{sec}\), and the sample standard deviation was \(.20 \mathrm{sec}\). Does it appear that true average task time is less than \(2 \mathrm{sec}\) ? Carry out a test of appropriate hypotheses using a significance level of .01.
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