Chapter 11

In a survey, \(8 \%\) of adult Internet users said they participate in sports fantasy leagues online. The margin of error is \(\pm 4 \%\). Describe and correct the error in calculating the sample size. $$ \begin{aligned} \pm 0.08 &=\pm \frac{1}{\sqrt{n}} \\ 0.0064 &=\frac{1}{n} \\ n & \approx 156 \end{aligned} $$

The staff of a student newsletter wants to conduct a survey of the students' favorite television shows. There are 1225 students in the school. Describe a method for selecting a random sample of 250 students to survey.

REASONING Will replicating an experiment on many individuals produce data that are more likely to accurately represent a population than performing the experiment only once? Explain.

Determine whether the inverse of \(f\) is a function. Then find the inverse. \(f(x)=\frac{3}{x+5}\)

In a random sample of 2500 consumers, \(61 \%\) prefer Game A over Game B. Describe and correct the error in giving an interval that is likely to contain the exact percent of all consumers who prefer Game A over Game B. Margin of error \(=\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{2500}}=0.02\) It is likely that the exact percent of all consumers who prefer Game A over Game B is between \(60 \%\) and \(62 \%\).

A national collegiate athletic association wants to survey 15 of the 120 head football coaches in a division about a proposed rules change. Describe a method for selecting a random sample of coaches to survey.

Ages: \(24,21,22,26,22,23,25,23,23,24,20,25\)

Determine whether the inverse of \(f\) is a function. Then find the inverse. \(f(x)=\frac{1}{2 x-1}\)

MAKING AN ARGUMENT A random sample of five people at a movie theater from a population of 200 people gave the film 4 out of 4 stars. Your friend concludes that everyone in the movie theater would give the film 4 stars. Is your friend correct? Explain your reasoning.

Your friend states that it is possible to have a margin of error between 0 and 100 percent, not including 0 or 100 percent. Is your friend correct? Explain your reasoning.

Golf strokes: \(4,3,4,3,3,2,7,5,3,4\)

Determine whether the inverse of \(f\) is a function. Then find the inverse. \(f(x)=\frac{2}{x}-4\)

In Exercises 21–24, identify the method of data collection the situation describes. The owner of a restaurant asks 20 customers whether they are satisfied with the quality of their meals.

Determine whether the inverse of \(f\) is a function. Then find the inverse. \(f(x)=\frac{3}{x^2}+1\)

OPEN-ENDED Find a newspaper or magazine article that describes a survey. Identify the population and sample. Describe the sample.

In Exercises 21–24, identify the method of data collection the situation describes. A researcher compares incomes of people who live in rural areas with those who live in large urban areas.

\(y=(0.95)^x\)

THOUGHT PROVOKING You choose a random sample of 200 from a population of 2000 . Each person in the sample is asked how many hours of sleep he or she gets each night. The mean of your sample is 8 hours. Is it possible that the mean of the entire population is only \(7.5\) hours of sleep each night? Explain.

Suppose a random sample of size \(n\) is required to produce a margin of error of \(\pm E\). Write an expression in terms of \(n\) for the sample size needed to reduce the margin of error to \(\pm \frac{1}{2} E\). How many times must the sample size be increased to cut the margin of error in half? Explain.

In Exercises 21–24, identify the method of data collection the situation describes. A researcher places bacteria samples in two different climates. The researcher then measures the bacteria growth in each sample after 3 days.

\(y=(0.2)^x\)

\(y=(1.25)^x\)

PROBLEM SOLVING You roll an eight-sided die five times and get a four every time. You suspect that the die favors the number four. The die maker claims that the die does not favor the number four. a. Perform a simulation involving 50 trials of rolling the actual die and getting a four to test the die maker's claim. Display the results in a histogram. b. What should you conclude when you roll the actual die 50 times and get 20 fours? 7 fours?

Consider a large population in which \(\rho\) percent (in decimal form) have a certain characteristic. To be reasonably sure that you are choosing a sample that is representative of a population, you should choose a random sample of \(n\) people where $$ n>9\left(\frac{1-\rho}{\rho}\right) \text {. } $$ a. Suppose \(\rho=0.5\). How large does \(n\) need to be? b. Suppose \(\rho=0.01\). How large does \(n\) need to be? c. What can you conclude from parts (a) and (b)?

\(x^2-10 x-4=0\)

In Exercises 25–28, explain why the survey question may be biased or otherwise introduce bias into the survey. Then describe a way to correct the flaw. "The tap water coming from our western water supply contains twice the level of arsenic of water from our eastern supply. Do you think the government should address this health problem?"

\(3 t^2+6 t=18\)

Find the inverse of the function. $$ y=10^{x-3} $$

In Exercises 25–28, explain why the survey question may be biased or otherwise introduce bias into the survey. Then describe a way to correct the flaw. A child asks, "Do you support the construction of a new children's hospital?"

\(s^2+10 s+8=0\)

Find the inverse of the function. $$ y=2^x-5 $$

In Exercises 29–32, determine whether the survey question may be biased or otherwise introduce bias into the survey. Explain your reasoning. "Do you favor government funding to help prevent acid rain?"

\(n^2+2 n+2=0\)

Find the inverse of the function. $$ y=\ln (x+5) $$

In Exercises 29–32, determine whether the survey question may be biased or otherwise introduce bias into the survey. Explain your reasoning. "Do you think that renovating the old town hall would be a mistake?"

\(4 z^2+28 z=15\)

Find the inverse of the function. $$ y=\log _6 x-1 $$

In Exercises 29–32, determine whether the survey question may be biased or otherwise introduce bias into the survey. Explain your reasoning. A police officer asks mall visitors, "Do you wear your seat belt regularly?"

.\(5 w-w^2=-11\)

A researcher studies the effect of fiber supplements on heart disease. The researcher identified 175 people who take fiber supplements and 175 people who do not take fiber supplements. The study found that those who took the supplements had \(19.6 \%\) fewer heart attacks. The researcher concludes that taking fiber supplements reduces the chance of heart attacks. a. Explain why the researcher's conclusion may not be valid. b. Describe how the researcher could have conducted the study differently to produce valid results.

What is the difference between a "blind experiment" and a "double-blind experiment?' Describe a possible advantage of the second type of experiment over the first.

A college wants to survey its graduating seniors to find out how many have already found jobs in their field of study after graduation. a. What is the objective of the survey? b. Describe the population for the survey. c. Write two unbiased questions for the survey.

About \(3.2 \%\) of U.S. adults follow a vegetarian-based diet. Two randomly selected groups of people were asked whether they follow such a diet. The first sample consists of 20 people and the second sample consists of 200 people. Which sample proportion is more likely to be representative of the national percentage? Explain.

The U.S. Census is taken every 10 years to gather data from the population. Your friend claims that the sample cannot be biased. Is your friend correct? Explain.

An airline wants to know whether travelers have enough leg room on its planes. a. What method of data collection is appropriate for this situation? b. Describe a sampling method that is likely to give biased results. Explain. c. Describe a sampling method that is not likely to give biased results. Explain. d. Write one biased question and one unbiased question for this situation.

A website contains a link to a survey that asks how much time each person spends on the Internet each week. a. What type of sampling method is used in this situation? b. Which population is likely to respond to the survey? What can you conclude?

Evaluate the expression without using a calculator. $$ 4^{5 / 2} $$

Evaluate the expression without using a calculator. $$ 27^{2 / 3} $$

Evaluate the expression without using a calculator. $$ -64^{1 / 3} $$

Evaluate the expression without using a calculator. $$ 8^{-2 / 3} $$