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

drinking water should contain at most $1.3$mg/liter of copper, on average. A water supply company is testing water from a new source and collects water in small bottles at each of$30$randomly selected locations. The company performs a test at the α=0.05 significance level of$H_0:\mathrm{μ}=1.3$versus $H_a:\mathrm{μ}>1.3$, 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 $\mathrm{Î\pm }=0.05$? Why or why not?

More lefties?In the population of people in the United States, about 10% are left-handed. After bumping elbows at lunch with several left-handed students, Simon wondered if more than $10\%$of students at his school are left-handed. To investigate, he selected an SRS of $50$students and found $8$lefties $(\stackrel{Áåœ}{p}=8/50=0.16)$.

To determine if these data provide convincing evidence that more than $10\%$of the students at Simon’s school are left-handed, $200$trials of a simulation were conducted. Each dot in the graph shows the proportion of students that are left-handed in a random sample of $50$students, assuming that each student has a $10\%$chance of being left handed.

a. State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

b. Use the simulation results to estimate the P-value of the test in part (a). Interpret the $P$-value.

c. What conclusion would you make?

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a loud noise as stimulus. The appropriate hypotheses for the significance test are

a. $H_0:\mathrm{μ}=18;H_a:\mathrm{μ}≠18$

b. $H_0:\mathrm{μ}=18;H_a:\mathrm{μ}>18$

c. $H_0:\mathrm{μ}<18;H_a:\mathrm{μ}=18$

d. $H_0:\mathrm{μ}=18;H_a:\mathrm{μ}<18$

e. $H_0:\stackrel{¯}{x}=18;H_a:\stackrel{¯}{x}<18$

How much juice? One company's bottles of grapefruit juice are filled by a machine that is set to dispense an average of $180$milliliters (ml) of liquid. A quality-control inspector must check that the machine is working properly. The inspector takes a random sample of $40$bottles and measures the volume of liquid in each bottle.

state appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest

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 $300$city residents and use the results to test the following hypotheses:

$H_0:p=0.50$

$H_a:p>0.50$

where $p$is 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.

$18\%$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 $300$city residents and use the results to test the following hypotheses:

$H_0:p=0.50$

$H_a:p>0.50$

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

In the sample, $\hat{p}=158/300=0.527$, The resulting $P$-value is $0.18$. What is the correct interpretation of this $P$-value?

a. Only $18\%$ of the city residents support the tax increase.

b. There is an $18\%$chance that the majority of residents supports the tax increase.

c. Assuming that $50\%$of residents support the tax increase, there is an $18\%$probability that the sample proportion would be $0.527$or greater by chance alone.

d. Assuming that more than $50\%$of residents support the tax increase, there is an $18\%$probability that the sample proportion would be $0.527$or greater by chance alone.

e. Assuming that $50\%$of residents support the tax increase, there is an $18\%$ chance that the null hypothesis is true by chance alone.

Based on the $P$-value in Exercise 31, which of the following would be the most

appropriate conclusion?

a. Because the P-value is large, we reject $H_0$. We have convincing evidence that more than $50\%$of city residents support the tax increase.

b. Because the $P$-value is large, we fail to reject $H_0$. We have convincing evidence that more than $50\%$of city residents support the tax increase.

c. Because the $P$-value is large, we reject $H_0$. We have convincing evidence that at most $50\%$of city residents support the tax increase.

d. Because the $P$-value is large, we fail to reject $H_0$. We have convincing evidence that at most $50\%$of city residents support the tax increase.

e. Because the $P$-value is large, we fail to reject $H_0$. We do not have convincing

evidence that more than $50\%$of city residents support the tax increase.

Explaining confidence: Here is an explanation from a newspaper concerning one of its opinion polls. Explain what is wrong with the following statement.

For a poll of $1600$ adults, the variation due to sampling error is no more than $3$

percentage points either way. The error margin is said to be valid at the $95\%$

confidence level. This means that, if the same questions were repeated in $20$ polls, the results of at least $19$ surveys would be within $3$ percentage points of the results of this survey.

Home computers Jason reads a report that says $80\%$of U.S. high school

students have a computer at home. He believes the proportion is smaller than $0.80$at his large rural high school. Jason chooses an SRS of $60$students and finds that $41$have a computer at home. He would like to carry out a test at the $\mathrm{Î\pm }=0.05$significance level of $H_0:p=0.80$versus $H_a:p<0.80$, where $p$= the true

proportion of all students at Jason’s high school who have a computer at home. Check if the conditions for performing the significance test are met.

Walking to school A recent report claimed that $13\%$of students typically walk to school. DeAnna thinks that the proportion is higher than $0.13$at her large elementary school. She surveys a random sample of $100$students and finds that $17$typically walk to school. DeAnna would like to carry out a test at the $\mathrm{Î\pm }=0.05$significance level of $H_0:p=0.13$versus $H_a:p>0.13$, where $p$= the true proportion of all students at her elementary school who typically walk to school. Check if the conditions for performing the significance test are met.