Chapter 11

A method in which new samples are repeatedly drawn from the data set is called ____.

COMPLETE THE SENTENCE A portion of a population that can be studied in order to make predictions about the entire population is \(\mathrm{a}(\mathrm{n})\)

The ___________ gives a limit on how much the responses of the sample would differ from the responses of the population.

Describe the difference between a stratifi ed sample and a cluster sample.

COMPLETE THE SENTENCE Repetition of an experiment under the same or similar conditions is called

WRITING Describe the difference between a parameter and a statistic. Give an example of each.

What is the difference between descriptive and inferential statistics?

A sample for which each member of a population has an equal chance of being selected is a(n) __________ sample.

WRITING Describe the difference between the control group and the treatment group in a controlled experiment

VOCABULARY What is a hypothesis in statistics?

The numbers of text messages sent each day by a random sample of 30 teen cellphone users are shown in the table. Estimate the population mean \(\mu\). Number of Text Messages $$ \begin{array}{|l|l|l|l|l|} \hline 30 & 60 & 59 & 83 & 41 \\ 37 & 66 & 63 & 60 & 92 \\ 53 & 42 & 47 & 32 & 79 \\ 53 & 80 & 41 & 51 & 85 \\ 73 & 71 & 69 & 31 & 69 \\ 57 & 60 & 70 & 91 & 67 \\ \hline \end{array} $$

Describe a situation in which you would use a simulation to collect data.

Insomnia New Drug Improves Sleep To test a new drug for insomnia, a pharmaceutical company randomly divided 200 adult volunteers into two groups. One group received the drug and one group received a placebo. After one month, the adults who took the drug slept \(18 \%\) longer, while those who took the placebo experienced no significant change.

WRITING Describe two ways you can make an incorrect decision when analyzing a hypothesis.

The incomes for a random sample of 35 U.S. households are shown in the table. Estimate the population mean \(\mu\). Income of U.S. Households $$ \begin{array}{|l|l|l|l|l|} \hline 14,300 & 52,100 & 74,800 & 51,000 & 91,500 \\ 72,800 & 50,500 & 15,000 & 37,600 & 22,100 \\ 40,000 & 65,400 & 50,000 & 81,100 & 99,800 \\ 43,300 & 32,500 & 76,300 & 83,400 & 24,600 \\ 30,800 & 62,100 & 32,800 & 21,900 & 64,400 \\ 73,100 & 20,000 & 49,700 & 71,000 & 45,900 \\ 53,200 & 45,500 & 55,300 & 19,100 & 63,100 \\ \hline \end{array} $$

Describe the difference between an unbiased sample and a biased sample. Give one example of each.

Dental Health Milk Fights Cavities At a middle school, students can choose to drink milk or other beverages at lunch. Seventy-five students who chose milk were monitored for one year, as were 75 students who chose other beverages. At the end of the year, students in the "milk" group had \(25 \%\) fewer cavities than students in the other group.

In Exercises 5–8, identify the type of sample described. The owners of a chain of 260 retail stores want to assess employee job satisfaction. Employees from 12 stores near the headquarters are surveyed.

In Exercise 4, interpret the meaning of \(\bar{x}_{\text {treatment }}-\bar{x}_{\text {control }}\) when the difference is positive, negative, and zero.

A survey asks a random sample of U.S. teenagers how many hours of television they watch each night. The survey reveals that the sample mean is 3 hours per night. How confident are you that the average of all U.S. teenagers is exactly 3 hours per night? Explain your reasoning.

In Exercises 5–8, identify the type of sample described. Each employee in a company writes their name on a card and places it in a hat. The employees whose names are on the first two cards drawn each win a gift card.

When the President of the United States vetoes a bill, the Congress can override the veto by a two-thirds majority vote in each House. Five news organizations conduct individual random surveys of U.S. Senators. The senators are asked whether they will vote to override the veto. The results are shown in the table. (See Example 2.) $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Sample } \\ \text { Size } \end{array} & \begin{array}{c} \text { Number of Votes } \\ \text { to Override Veto } \end{array} & \begin{array}{c} \text { Percent of Votes } \\ \text { to Override Veto } \end{array} \\ \hline 7 & 6 & 85.7 \% \\ 22 & 16 & 72.7 \% \\ 28 & 21 & 75 \% \\ 31 & 17 & 54.8 \% \\ 49 & 27 & 55.1 \% \\ \hline \end{array} $$ a. Based on the results of the first two surveys, do you think the Senate will vote to override the veto? Explain. b. Based on the results in the table, do you think the Senate will vote to override the veto? Explain.

In Exercises 5–8, identify the type of sample described. A taxicab company wants to know whether its customers are satisfied with the service. Drivers survey every tenth customer during the day.

Your teacher lets the students decide whether to have their test on Friday or Monday. The table shows the results from four surveys of randomly selected students in your grade who are taking the same class. The students are asked whether they want to have the test on Friday. $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Sample } \\ \text { Size } \end{array} & \begin{array}{c} \text { Number of } \\ \text { "Yes" Responses } \end{array} & \begin{array}{c} \text { Percent of } \\ \text { Votes } \end{array} \\ \hline 10 & 8 & 80 \% \\ 20 & 12 & 60 \% \\ 30 & 16 & 53.3 \% \\ 40 & 18 & 45 \% \\ \hline \end{array} $$ a. Based on the results of the first two surveys, do you think the test will be on Friday? Explain. b. Based on the results in the table, do you think the test will be on Friday? Explain.

In Exercises 5–8, identify the type of sample described. The owner of a community pool wants to ask patrons whether they think the water should be colder. Patrons are divided into four age groups, and a sample is randomly surveyed from each age group.

A national polling company claims that \(54 \%\) of U.S. adults are married. You survey a random sample of 50 adults. (See Example 3.) a. What can you conclude about the accuracy of the claim that the population proportion is \(0.54\) when 31 adults in your survey are married? b. What can you conclude about the accuracy of the claim that the population proportion is \(0.54\) when 19 adults in your survey are married? c. Assume that the true population proportion is 0.54. Estimate the variation among sample proportions for samples of size 50.

In the United States, a survey of 1777 adults ages 18 and over found that 1279 of them do some kind of spring cleaning every year.

Employee engagement is the level of commitment and involvement an employee has toward the company and its values. A national polling company claims that only \(29 \%\) of U.S. employees feel engaged at work. You survey a random sample of 50 U.S. employees. a. What can you conclude about the accuracy of the claim that the population proportion is \(0.29\) when 16 employees feel engaged at work? b. What can you conclude about the accuracy of the claim that the population proportion is \(0.29\) when 23 employees feel engaged at work? c. Assume that the true population proportion is \(0.29\). Estimate the variation among sample proportions for samples of size 50 .

In Exercises 9–12, identify the type of sample and explain why the sample is biased. A sportswriter wants to determine whether baseball coaches think wooden bats should be mandatory in collegiate baseball. The sportswriter mails surveys to all collegiate coaches and uses the surveys that are returned.

In a school district, a survey of 1300 high school students found that 1001 of them like the new, healthy cafeteria food choices.

In Exercises 9–12, identify the type of sample and explain why the sample is biased. You want to find out whether booth holders at a convention were pleased with their booth locations. You divide the convention center into six sections and survey every booth holder in the fifth section.

DRAWING CONCLUSIONS A company wants to test whether a nutritional supplement has an adverse effect on an athlete’s heart rate while exercising. Identify a potential problem, if any, with each experimental design. Then describe how you can improve it. (See Example 3.) a. The company randomly selects 250 athletes. Half of the athletes receive the supplement and their heart rates are monitored while they run on a treadmill. The other half of the athletes are given a placebo and their heart rates are monitored while they lift weights. The heart rates of the athletes who took the supplement signifi cantly increased while exercising. b. The company selects 1000 athletes. The athletes are divided into two groups based on age. Within each age group, the athletes are randomly assigned to receive the supplement or the placebo. The athletes’ heart rates are monitored while they run on a treadmill. There was no signifi cant difference in the increases in heart rates between the two groups.

In Exercises 9–12, identify the type of sample and explain why the sample is biased. Every tenth employee who arrives at a company health fair answers a survey that asks for opinions about new health-related programs.

2\. DRAWING CONCLUSIONS A researcher wants to test the effectiveness of reading novels on raising intelligence quotient (IQ) scores. Identify a potential problem, if any, with each experimental design. Then describe how you can improve it. a. The researcher selects 500 adults and randomly divides them into two groups. One group reads novels daily and one group does not read novels. At the end of 1 year, each adult is evaluated and it is determined that neither group had an increase in IQ scores. b. Fifty adults volunteer to spend time reading novels every day for 1 year. Fifty other adults volunteer to refrain from reading novels for 1 year. Each adult is evaluated and it is determined that the adults who read novels raised their IQ scores by 3 points more than the other group

Your friend states that the mean of the resampling differences of the means should be close to 0 as the number of resamplings increase. Is your friend correct? Explain your reasoning.

Surveys are mailed to every other household in a neighborhood. Each survey that is returned is used. Describe and correct the error in identifying the type of sample that is used. Because the surveys were mailed to every other household, the sample is a systematic sample.

Describe an example of an observation that can be made from an experiment. Then give four possible inferences that could be made from the observation.

A researcher wants to know whether the U.S. workforce supports raising the minimum wage. Fifty high school students chosen at random are surveyed. Describe and correct the error in determining whether the sample is biased. Because the students were chosen at random, the sample is not biased.

WRITING Explain why observational studies, rather than experiments, are usually used in astronomy

MAKING AN ARGUMENT Your friend wants to determine whether the number of siblings has an effect on a student’s grades. Your friend claims to be able to show causality between the number of siblings and grades. Is your friend correct? Explain.

Factor the polynomial completely. \(5 x^3-15 x^2\)

Factor the polynomial completely. \(y^3-8\)

ERROR ANALYSIS A survey of 1270 high school students found that 965 students felt added stress because of their workload. Describe and correct the error in identifying the population and the sample.

In Exercises 15–18, determine whether the sample is biased. Explain your reasoning. To assess customers' experiences making purchases online, a rating company e-mails purchasers and asks that they click on a link and complete a survey.

WRITING Describe the placebo effect and how it affects the results of an experiment. Explain how a researcher can minimize the placebo effect.

Factor the polynomial completely. \(z^3+5 z^2-9 z-45\)

ERROR ANALYSIS Of all the players on a National Football League team, the mean age is 26 years. Describe and correct the error in determining whether the mean age represents a parameter or statistic.

In Exercises 15–18, determine whether the sample is biased. Explain your reasoning. Your school principal randomly selects fi ve students from each grade to complete a survey about classroom participation.

THOUGHT PROVOKING Make a hypothesis about something that interests you. Design an experiment that could show that your hypothesis is probably true.

Factor the polynomial completely. \(81 w^4-16\)