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Chapter 10: Problem 18

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 \(150^{\circ} \mathrm{F}\), researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ} \mathrm{F}\) versus \(H_{a}: \mu>150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

Short Answer

Expert verified
A Type I error would be to incorrectly find that the mean temperature of the discharged water is more than 150掳F, while a Type II error would be to not detect that the mean water temperature is more than 150掳F when it actually is. Given the potential severe negative impact on the river's ecosystem associated with a Type II error, it would be considered more serious in this context.

Step by step solution

01

Understanding Type I and Type II errors

In statistical hypothesis testing, a Type I error occurs when the null hypothesis is rejected when it is true, while a Type II error occurs when the null hypothesis is not rejected when it is false. In this case, Type I error would mean incorrectly concluding that the temperature is more than 150 degrees when it is not, and Type II error would mean incorrectly concluding that the temperature is not more than 150 degrees when it really is.
02

Analyzing the implications of Type I and Type II errors

A Type I error in this scenario would raise unnecessary alarms as power plant's discharge water would be falsely marked as a threat to the river's ecosystem. On the other hand, a Type II error means that the power plant's discharge water's harmful effects are being overlooked which can result in severe damage to river's ecosystem.
03

Determine which type of error is more serious

In this context, a Type II error is more serious because it means the power plant's discharge water's temperature is more than 150 degrees and it is going unnoticed. This could lead to severe damage to the river's ecosystem, which is a much higher cost than raising false alarms (Type I error).

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Most popular questions from this chapter

A certain university has decided to introduce the use of plus and minus with letter grades, as long as there is evidence that more than \(60 \%\) of the faculty favor the change. A random sample of faculty will be selected, and the resulting data will be used to test the relevant hypothe ses. If \(\pi\) represents the true proportion of all faculty that favor a change to plus- minus grading, which of the following pair of hypotheses should the administration test: $$ H_{0}: \pi=.6 \text { versus } H_{a}: \pi<.6 $$ or $$ H_{0}: \pi=.6 \text { versus } H_{a}: \pi>.6 $$ Explain your choice.

In a survey conducted by Yahoo Small Business. 1432 of 1813 adults surveyed said that they would alter their shopping habits if gas prices remain high (Associated Press, November 30,2005 ). The article did not say how the sample was selected, but for purposes of this exercise, assume that it is reasonable to regard this sample as representative of adult Americans. Based on these survey data, is it reasonable to conclude that more than three-quarters of adult Americans plan to alter their shopping habits if gas prices remain high?

The article "Fewer Parolees Land Back Behind Bars" (Associated Press, April 11,2006 ) includes the following statement: "Just over 38 percent of all felons who were released from prison in 2003 landed back behind bars by the end of the following year, the lowest rate since 1979." Explain why it would not be necessary to carry out a hypothesis test to determine if the proportion of felons released in 2003 was less than \(.40\).

Pizza Hut, after test-marketing a new product called the Bigfoot Pizza, concluded that introduction of the Bigfoot nationwide would increase its sales by more than \(14 \%\) (USA Today, April 2, 1993). This conclusion was based on recording sales information for a random sample of Pizza Hut restaurants selected for the marketing trial. With \(\mu\) denoting the mean percentage increase in sales for all Pizza Hut restaurants, consider using the sample data to decide between \(H_{0}: \mu=14\) and \(H_{a}: \mu>14\). a. Is Pizza Hut's conclusion consistent with a decision to reject \(H_{0}\) or to fail to reject \(H_{0}\) ? b. If Pizza Hut is incorrect in its conclusion, is the company making a Type I or a Type II error?

A hot tub manufacturer advertises that with its heating equipment, a temperature of \(100^{\circ} \mathrm{F}\) can be achieved in at most \(15 \mathrm{~min}\). A random sample of 25 tubs is selected, and the time necessary to achieve a \(100^{\circ} \mathrm{F}\) temperature is determined for each tub. The sample average time and sample standard deviation are \(17.5\) min and \(2.2\) min, respectively. Does this information cast doubt on the company's claim? Carry out a test of hypotheses using significance level \(.05 .\)
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