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Chapter 9: Q 26. (page 566)

Clean water The Environmental Protection Agency (EPA) has determined that safe

drinking water should contain at most 1.3mg/liter of copper, on average. A water supply company is testing water from a new source and collects water in small bottles at each of30randomly selected locations. The company performs a test at the α=0.05 significance level ofH0:μ=1.3versus Ha:μ>1.3, where μ is the

true mean copper content of the water from the new source.

a. Describe a Type I error and a Type II error in this setting.

b. Which type of error is more serious in this case? Justify your answer.

c. Based on your answer to part (b), do you agree with the company’s choice of α=0.05? Why or why not?

Short Answer

Expert verified

a. Type I error rejects null hypothesis H0if it is true and type II error fails to reject null hypothesis H0if it is false.

b. Type II error is more serious.

c.α=0.05is better thanα=0.10

Step by step solution

01

Given Information

It is given that H0:μ=1.3

H1:μ>1.3

02

Explaining type I and type II error

Type I error: There is enough convincing evidence that true mean copper content from new source is 1.3mg/l, when actual mean content of water from new source is 1.3mg/l.

Type II error: Noconvincing evidence is present true mean copper content from new source is 1.3mg/l when actual mean content of water from new source is over1.3 mg/l

03

Which error is worse

Here, type II error is worse as when water is not safe, people might drink it.

04

Choice of α

As type II error is more worse.

The chances of type II error decreases with increase in type I error and αwould be better.

Hence, α=0.10is preferred over0.05

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Most popular questions from this chapter

Stating hypotheses

a. A change is made that should improve student satisfaction with the parking situation at your school. Before the change, 37%of students approve of the parking that's provided. The null hypothesis H0:p∧=0.37H0:p^=0.37is tested against the alternative Ha: p∧>0.37Ha:p^>0.37

b. A researcher suspects that the mean birth weights of babies whose mothers did not see a doctor before delivery is less than 3000 grams. The researcher states the hypotheses as

H0:μ=3000gramsH0:μ=3000grams

Ha:μ≤2999gramsHa:μ≤2999grams

Which of the following 95%confidence intervals would lead us to reject H0:p=0.30in favor of Ha:pnotequalto0.30at the 5%significance level?

a. (0.19,0.27)

b.(0.24,0.30)

c. (0.27,0.31)

d. (0.29,0.31)

e. None of these

Jump around Student researchers Haley, Jeff, and Nathan saw an article on the Internet claiming that the average vertical jump for teens was 15 inches. They wondered if the average vertical jump of students at their school differed from 15 inches, so they obtained a list of student names and selected a random sample of 20 students. After contacting these students several times, they finally convinced them to allow their vertical jumps to be measured. Here are the data (in inches):

Do these data provide convincing evidence at the α=0.10 level that the average vertical jump of students at this school differs from 15 inches?

Members of the city council want to know if a majority of city residents supports a 1%increase in the sales tax to fund road repairs. To investigate, they survey a random sample of 300city residents and use the results to test the following hypotheses:

H0:p=0.50

Ha:p>0.50

where pis the proportion of all city residents who support a 1% increase in the sales tax to fund road repairs.

A Type I error in the context of this study occurs if the city council

a. finds convincing evidence that a majority of residents supports the tax increase, when in reality there isn’t convincing evidence that a majority supports the increase.

b. finds convincing evidence that a majority of residents supports the tax increase, when in reality at most 50%of city residents support the increase.

c. finds convincing evidence that a majority of residents supports the tax increase, when in reality more than 50%of city residents do support the increase.

d. does not find convincing evidence that a majority of residents supports the tax increase, when in reality more than 50%of city residents do support the increase.

Side effects A drug manufacturer claims that less than 10%of patients who take its new drug for treating Alzheimer’s disease will experience nausea. To test this claim, researchers conduct an experiment. They give the new drug to a random sample of 300out of 5000Alzheimer’s patients whose families have given informed consent for the patients to participate in the study. In all, 25of the subjects experience nausea.

a. Describe a Type I error and a Type II error in this setting, and give a possible

consequence of each.

b. Do these data provide convincing evidence for the drug manufacturer’s claim?
See all solutions

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