math 10-12
Many states run lotteries to raise money. A website advertises that it knows "how to increase YOUR chances of Winning the Lottery." They offer several systems and criticize others as foolish. One system is called Lucky Numbers. People who play the Lucky Numbers system just pick a "lucky" number to play, but maybe some numbers are luckier than others. Let's use a simulation to see how well this system works. To make the situation manageable, simulate a simple lottery in which a single digit from 0 to 9 is selected as the winning number. Pick a single value to bet, such as 1, and keep playing it over and over. You'll want to run at least 100 trials. a) What proportion of the time do you expect to win? b) Would you expect better results if you picked a "luckier" number, such as 7? (Try it if you don't know.) Question content area bottom Part 1
About 10% of the time no
Which of the following is a characteristic of a good experiment?
All of the above
For the following report about a statistical study, identify (if possible) a) the population; b) the population parameter of interest; c) the sampling frame; d) the sample; e) the sampling method, including whether or not randomization was employed; f) who (if anyone) was left out of the study; and g) any potential sources of bias you can detect and any problems you see in generalizing to the population of interest. Hoping to learn what issues may resonate with voters in the coming election, the campaign director for a mayoral candidate randomly selects six blocks from each of the city's election districts. Staff members go there and interview all the residents they can find. The residents were asked to select the five most important issues from a prepared list.
All voters in the city The most important issues to voters All city residents Stratified sample, with each district being represented by the six blocks Multistage sample, using stratified sample with districts as strata and cluster sample within each district yes The staff members interviewed "all the residents they can find," so anyone they could not find was left out of the study. Undercoverage; staff members interviewed "all the residents they can find" and those residents may have different priorities than those who cannot be found.
The 1990s and early 2000s could be considered the steroids era in Major League Baseball, as many players have admitted to using the drug to increase performance on the field. A sports writer wanted to compare home run totals from the steroids era to the 1960s. Complete parts a and b.
An observational study, because the sports writer is not randomly assigning players to take steroids or not take steroids; the writer is merely observing home run totals between two eras. It would be unwise to conclude steroids caused an increase in home runs because causation cannot be concluded from an observational study. Other variables could include diet and exercises
Between quarterly audits, a company likes to check on its accounting procedures to address any problems before they become serious. The accounting staff processes payments on about 120 orders each day. The next day, the supervisor rechecks 10 of the transactions to be sure they were processed properly. Complete parts a and b below.
Assign numbers 001 to 120 to each order. Use random numbers to select 10 transactions to examine. Sample proportionately within each type before the results are combined. This is a stratified random sample.
At its website, the Gallup World Poll describes its methods. After one report it explained: Results are based on face-to-face interviews with randomly selected national samples of approximately 1,000 adults, aged 15 and older, who live permanently in each of the 21 sub-Saharan African nations surveyed. Those countries include Angola (areas where land mines might be expected were excluded), Benin, Botswana, Burkina Faso, Cameroon, Ethiopia, Ghana, Kenya, Madagascar (areas where interviewers had to walk more than 20 kilometers from a road were excluded), Mali, Mozambique, Niger, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda (the area of activity of the Lord's Resistance Army was excluded from the survey), Zambia, and Zimbabwe. . . . In all countries except Angola, Madagascar, and Uganda, the sample is representative of the entire population. Use this to complete parts a and b.
Multistage sampling The difference in population size has no effect whatever on the precision of estimates from these surveys. Only the sample size matters.
A question posted on a video game website asked visitors to the site, "Do you have an active social life outside the Internet?" 31% of the 34,923 respondents said "No" or "Not really, most of my personal contact is online." a) Can this survey be used to estimate the proportion of adults who would say they have an active social life outside the Internet? Why or why not? b) Can this survey be used to estimate the proportion of visitors to the site who would say they have an active social life outside the Internet? Why or why not?
No, because the people who visit the site would not be representative of the population of all adults. No, because the survey is a voluntary response, those who answer are more likely to have stronger feelings about the question.
a friend asks you to quickly name a professional sports team. is the sports team random?
No, since most people have a biased knowledge of professional sports teams
A basketball player takes a foul shot. Look at a random digit, using an odd digit to represent a good shot and an even digit to represent a miss.
The even-odd assignment assumes that the player is equally likely to make a good shot or miss the shot. In reality, the likelihood of making a good shot depends on the player's skill.
Simulate a quarterback's performance by letting 0 be an incomplete pass, 1 be a complete pass, and 2 be an interception.
The likelihoods for an incomplete pass comma a complete pass comma and an interception are not the same comma but the simulation assumes they are.
A group of students in your intro stats class design an experiment to test whether popcorn stored in the freezer pops better (fewer kernels left un-popped) than room temperature popcorn. In addition, they also want to test different power levels on their microwaves. The factor(s) in this experiment is(are) ___________________.
are temperature and power.
A friend of yours is a very busy woman. If you call her to study for class in the evening, she does not answer the phone 60% of the time, she answers and says no 20% of the time, and she answers and says yes 20% of the time. You create a simulation model for the number of nights in a week the two of you get together to study for class, and run the simulation for 20 trials. A report on your results could include
the average number of nights you studied together per week
proud legislator claims that your state's new law against talking on a wireless phone while driving has reduced wireless phone use to less than 12% of all drivers. While waiting for your bus the next morning, you notice that 4 of the 10 people who drive by are using their wireless phones. Does this cast doubt on the legislator's figure of 12%? Use a simulation to estimate the likelihood of seeing at least 4 of 10 randomly selected drivers talking on their wireless phones if the actual rate of usage is 12%. Explain your conclusion clearly. For the simulation, include 10 trials and use the accompanying random number table, letting 00-11 represent a driver using a wireless phone.
10 does unlikly
Major League Baseball (MLB) tests players to see whether they are using performance-enhancing drugs. Officials select a team at random, and a drug-testing crew shows up unannounced to test all 40 players on the team. Each testing day can be considered a study of drug use in Major League Baseball. Complete parts a through c.
All MLB players A cluster sample Yes. An entire team can be sampled at once relatively easily, but a random sample of all players could not efficiently be drawn on the same day.
A group of students in your intro stats class design an experiment to test whether popcorn stored in the freezer pops better (fewer kernels left un-popped) than room temperature popcorn. Two group members do the shopping separately and realize too late that they've bought two different brands. If they ignore this and randomly assign a mix of bags to each treatment group, what principle of experiment design have they ignored?
Blocking
Anytime a survey is conducted, care must be taken to avoid undercoverage. Consider drawing a random sample only from cell phone numbers. Discuss the advantages and disadvantages of such a sampling method compared with surveying randomly generated telephone numbers from non-cell phone numbers. Do you think these advantages and disadvantages have changed over time? How do you expect they'll change in the future?
Cell phones are typically not shared, so each phone number identifies a single individual Cell phones are less expensive and more likely to be used by the general public than non-cell phones. This will help avoid an undercoverage bias. People tend to keep their cell phones when they move, so it may be impossible to identify local respondents by their area code or exchange. As people keep the cell phone numbers they had as children and then move to other places, the potential disadvantages may increase with time.
A pizza delivery driver, always trying to increase tips, runs an experiment on his next 40 deliveries. He flips a coin to decide whether or not to call a customer from his mobile phone when he is five minutes away, hoping this slight bump in customer service will lead to a slight bump in tips. After 40 deliveries, he will compare the average tip percentage between the customers he called and those he did not. Complete parts a through f.
Each of the 40 deliveries is an experimental unit. He has randomized the experiment by flipping a coin to decide whether or not to phone. calling/not calling whether or not he calls the customer tip percentage for each delivery Season, Delivery vehicle Yes, because the driver flipped a coin to determine whether or not to call. Yes, because the driver flipped a coin for 40 deliveries. Customers don't know about the experiment, so the customers are blind. That would make this a single-blind study. It can't be double-blind because the delivery driver must know whether or not he phones. The cost or size of a delivery may confound his results. Larger orders may generally tip a higher or lower percentage of the bill. Yes. Each driver is now a block. The experiment is randomized within each block. This is a good idea because some drivers might generally get higher tips than others, but the goal of the experiment is to study the effect of phone calls.
Some people who race greyhounds give the dogs large doses of vitamin C in the belief that the dogs will run faster. Investigators at the University of Florida tried three different diets in random order on each of five racing greyhounds. They were surprised to find that when the dogs ate high amounts of vitamin C they ran more slowly. Complete parts a through h.
Experiment Greyhounds diet, 3 There are 3 treatments. Speed Completely Randomized Completely Randomized Greyhounds who eat diets high in vitamin C seem to run more slowly.
State police set up a roadblock to estimate the percentage of cars with up-to-date registration, insurance, and safety-inspection stickers. It would be too inconvenient and costly to check every vehicle that passes through a checkpoint, so they decide to stop about 1/20 of the vehicles. Complete parts a) and b) below.
If the police used a list of registered automobiles, they would have to track down the owners, and unregistered vehicles would not be on the list. Roll a 20-sided die for each car that passes through the checkpoint. If they roll a 1, the car gets checked. Use a systematic sample, checking every 20th vehicle that passes through the checkpoint. Randomly select a car out of the first 20, then check every 20th vehicle after that.
Many kinds of games people play rely on randomness. Cite three different methods commonly used in the attempt to achieve this randomness, and discuss the effectiveness of each. Three methods that are commonly used to achieve randomness include, but are not limited to: picking a certain number and having someone guess the number ("Method 1"); shuffling a deck of cards ("Method 2"); and flipping a coin ("Method 3"). Rank the methods.
Method 3 > Method 2 > Method 1 Method 1 involves only mental components. Method 3 involves only physical components. Method 2 involves both mental and physical components.
A basketball player with a 69% shooting percentage has just made 6 shots in a row. The announcer says this player "is hot tonight! She's in the zone!" Assume the player takes about 20 shots per game. Is it unusual for her to make 6 or more shots in a row during a game?
No, because this occurs in about 52% of the games she plays.
Researchers who examined health records of thousands of males found that men who died of myocardial infarction (heart attack) tended to be shorter than men who did not. Complete parts a and b below.
No, this is a retrospective study previous conditions are determined. No, since there may be lurking variables, such as patient age or weight.
Through its Roper Reports Worldwide, GfK Roper conducts a global consumer survey to help multinational companies understand different consumer attitudes throughout the world. Within 30 countries, the researchers interview 1000 people aged 13-65. Their samples are designed so that they get 500 men and 500 women in each country. Complete parts a and b below.
No. It would be nearly impossible to get exactly 500 men and 500 women from every country by random chance. A stratified sample, stratified by whether the respondent is a man or a woman.
For their class project, a group of Business students decide to survey the student body to assess opinions about a proposed new student coffee shop to judge how successful it might be. Their sample of 120 contained 30 first-year students, 30 sophomores, 30 juniors, and 30 seniors. Complete parts (a) and (b) below.
No; it would be nearly impossible to get exactly 30from each class by random chance. A stratified sample
Administrators at Texas A&M University were interested in estimating the percentage of students who are the first in their family to go to college. The A&M student body has about 48,000 members. Complete parts a) through e) below.
Several terms are poorly defined. The survey needs to specify the meaning of "family" for this purpose and the meaning of "higher education". The term "seek" is also poorly defined, as it does not specify what qualifies as seeking more education. cluster sample stratified sample systematic sample This is not an SRS. Although each student has an equal chance to be in the survey, groups of friends that sit in the same row will either be all in or all out of the sample, so not all samples have the same chance of being selected. This would suffer from undercoverage. This would make the percentage lower, since students who are the first in their family to go to college may be more likely to lack internet access. This would suffer from voluntary response bias, since it only samples students who choose to visit the site. This would make the sample unrecoverable. This would suffer from undercoverage, since it only samples students passing through the student union. This may bias the sample. This would suffer from nonresponse bias, since it only samples students passing through the student union. This may bias the sample. This would be a convenience sample, since it only samples students that are easy to reach. While this may not affect the percentage, it would make the sample much less likely to be representative The proportion in the sample is a statistic. The proportion of all students is the parameter of interest. The statistic estimates the parameter, but it is not likely to be exactly the same.
For the following report about a statistical study, identify (if possible) a) the population; b) the population parameter of interest; c) the sampling frame; d) the sample; e) the sampling method, including whether or not randomization was employed; f) who (if anyone) was left out of the study; and g) any potential sources of bias you can detect and any problems you see in generalizing to the population of interest. A company packaging snack foods maintains quality control by randomly selecting 10 cases from each day's production and weighing the bags. Then they open five random bags from each case and inspect the contents.
Snack food bags produced by the company The population parameter of interest is unclear. All bags produced by the company each day The 50 total bags that are opened from the selected cases Multistage sampling was used. yes No group was left out of the study. There is no indication of bias.
Two members of the PTA committee have proposed the accompanying questions to ask in seeking parent's opinions. Question 1 is "Should elementary school-age children have to pass high-stakes tests in order to remain with their classmates?" and Question 2 is "Should schools and students be held accountable for meeting yearly learning goals by testing students before they advance to the next grade?" Complete parts a and b below.
The answers for these two questions will definitely differ. Question 1 will probably get many "No" answers, while Question 2 will get many "Yes" answers. This is an example of response bias. "Do you think standardized tests are appropriate for deciding whether a student should be promoted to the next grade?
A Statistics student properly simulated the length of checkout lines in a grocery store and then reported, "The average length of the line will be 3.2 people." What is wrong with this conclusion?
The conclusion should indicate that the simulation suggests that the average length of the line would be 3.2 people. Future results might not match the simulated results exactly.
After simulating the spread of a disease, a researcher wrote, "24% of the people contracted the disease." What should the correct conclusion be?
The correct conclusion should be "According to the simulation, approximately 24% of people will contract the disease."
A man is throwing darts. Look at a random digit, using an odd digit to represent hitting the target and an even digit to represent missing.
The even-odd assignment assumes that the man is equally likely to hit or miss the target.In reality, the likelihood of hitting the target depends on the man's skills.
Prior to a mayoral election, a newspaper conducted a poll. The paper surveyed a random sample of registered voters stratified by political party, age, gender, and area of residence. This poll predicted that Amabo would win the election with 52% of the vote. The newspaper was wrong: Amabo lost, getting only 46% of the vote. Do you think the newspaper's faulty prediction is more likely to be a result of bias or sampling error? Explain.
The faulty prediction was more likely to be the result of sampling error. The description of the sampling method suggests that samples will be representative of the voting population. However, random chance in selecting the individuals who were polled means that sample statistics will vary from the population parameter.
Use random numbers from 2 through 12 to represent the sum of the faces when two dice are rolled.
The numbers would represent the sums, but the sums are not all equally likely, as this simulation assumes.
Some people have been complaining that the children's playground at a municipal park is too small and is in need of repair. Managers of the park decide to survey city residents to see if they believe the playground should be rebuilt. Use this to answer parts a through c.
The percentage of city residents who think the playground should be rebuilt The sampling scheme suffers from voluntary response bias. Parents who feel strongly about the issue are morely likely to respond than those who do not. The sampling scheme suffers from undercoverage, because only people who come to the park to use the playground will respond. City residents who do not have children or who do not visit the park will not be included. One solution would be to conduct a systematic random sample at a variety of public places in the city.
Examine each of the following questions for possible bias. If you think the question is biased, indicate how and propose a better question.
The question is biased toward "no" because of the word "compelled." A better question may be "Should companies be responsible for any costs to restore the rainforest" The question is biased toward "no" because of the preamble "16-year-olds are old enough to drive." A better question may be "Do you think the gambling age should be set at 18?"
Which matters more about a sample you draw from a population?
The size of the sample.
The managers of a large company wished to know the percentage of employees who feel "extremely satisfied" to work there. The company has roughly 24,000 employees. They contacted a random sample of employees and asked them about their job satisfaction, obtaining 437 completed responses. Complete parts a through e.
The survey result is a statistic. It estimates the true proportion of satisfied workers but does not give the value precisely. The survey result is a statistic. It estimates the true proportion of satisfied workers but does not give the value precisely. stratified sample systematic sample multistage sample The survey is likely to be biased because employees will not want to express unhappiness in front of their supervisors or their coworkers. This represents a convenience sample and it would also suffer from voluntary response bias. This suffers from nonresponse bias because not everyone will respond.
Examine each of the following questions for possible bias. If you think the question is biased, indicate how and propose a better question. a) Do you think high school students should be required to wear uniforms? b) Given humanity's great tradition of exploration, do you favor continued funding for space flights?
There is no indication of bias. The question may be biased towards yes because of "great tradition." A better question would be "Do you favor continued funding for the space program?"
A local TV station conducted a "PulsePoll" about the upcoming mayoral election. Evening news viewers were invited to text in their votes, with the results to be announced on the late-night news. Based on the texts, the station predicted that Amabo would win the election with 52% of the vote. They were wrong: Amabo lost, getting only 46% of the vote. Do you think the station's faulty prediction is more likely to be a result of bias or sampling error? Explain.
The station's faulty prediction is a result of sampling error. The description of the sampling method suggests that samples should be representative of the viewers of the TV station. However, not every viewer will text in their vote for the mayoral election.
For your political science class, you'd like to take a survey from a sample of all the Protestant Church members in your region. A list of places of worship shows 24 Protestant churches in the area. Rather than try to obtain a list of all members of all these churches, you decide to pick 4 churches at random. For those churches, you will ask to get a list of all current members and contact all of them.
This is a cluster sampling design. One of the churches you picked may not be representative of all churches
A business student conjectures that the Internet caused companies to become more profitable, because many transactions previously handled "face-to-face" could now be completed online. The student compares earnings from a sample of companies from the 1980s to a sample from the 2000s. Complete parts a and b.
This is an observational study because the student is not randomly assigning companies to use or not use the Internet for business transactions. She cannot conclude the Internet was the cause because it could have been due to a number of factors, not specifically the Internet as a means for conducting business.
For his Statistics class experiment, research J. Gilbert decided to study how parents' income affects children's performance on standardized tests like SAT. He proposed to collect information from a random sample of test takers and examine the relationship between parental income and SAT score. a) Is this an experiment? If not, what kind of study is it? b) If there is a relationship between parental income and SAT score, why can't we conclude that differences in score are caused by differences in parental income?
This is not an experiment. There are no manipulated factors. This is an observational study. Causation cannot be concluded from observational data.
Another strategy for beating the lottery is the reverse of the system described below. Each time, bet the number that just turned up.
This method does about the same.
Many states run lotteries, giving away millions of dollars if your set of numbers matches the set of winning numbers. A machine pops up numbered balls to determine the winning set of numbers. Do you think this method guarantees randomness? Explain.
This method is random only if the balls generate numbers in equal frequencies.
Consider each of the situations below. Do you think the proposed sampling method is appropriate?
This sampling method is not appropriate. This method will probably result in undercoverage of those doctors who did not purchase a Yellow Pages ad. This sampling method is not appropriate. The sample will probably contain listings for only one or two types of businesses, resulting in undercoverage.
Some people play state-run lotteries by always playing the same favorite "lucky" numbers. Assuming that the lottery is truly random, is this strategy better, worse, or the same as choosing different numbers for each play? Explain.si
This strategy is the same as choosing different numbers for each play. If the lottery is random, it does not matter which numbers are chosen for each play; all are equally likely to win.
An online poll on a website used the wording shown below. 65% of 17,303 respondents said "yes." Comment on each of the statements about the poll given in parts a through d. A nationwide ban of the diet supplement ephedra went into effect recently. The herbal stimulant has been linked to 155 deaths and many more heart attacks and strokes. Ephedra manufacturer NVE Pharmaceuticals, claiming that the FDA lacked proof that ephedra is dangerous if used as directed, was denied a temporary restraining order on the ban by a federal judge. Do you think that ephedra should continue to be banned nationwide?
This survey suffers from voluntary response bias. Even a very large sample size won't make it representative. The wording does not appear to be biased because it states the facts and gives voice to both sides of the issue. The sampling frame consists of all people who visit this particular website and are willing to respond to the question. This is a true statement.
When spending large amounts to purchase advertising time, companies want to know what audience they'll reach. A poll asked randomly selected American adults whether they planned to watch the upcoming football championship game. Men and women were asked separately whether they were looking forward more to the football game or to watching the commercials. Among the men who planned on watching, 70% were watching for the game. Among women, 60% were looking forward primarily to the game. a) Was this a stratified sample or a blocked experiment? Explain. b) Was the design of the study appropriate for the advertisers' questions?
This was a stratified sample because no treatment was applied, so the poll was not an experiment Yes, because the results of the study will give a fairly accurate result regarding proportions of men and women who were looking forward more to watching the commercials.
For the following report about a statistical study, identify (if possible) a) the population; b) the population parameter of interest; c) the sampling frame; d) the sample; e) the sampling method, including whether or not randomization was employed; f) who (if anyone) was left out of the study; and g) any potential sources of bias you can detect and any problems you see in generalizing to the population of interest. A US consumer magazine asked all adult subscribers whether they had used experimental medical treatments and, if so, whether they had benefited from them. For almost all of the treatments, approximately 22% of those responding reported cures or substantial improvement in their condition.
Unclear, but probably all adults in the country The proportion who have used and benefited from experimental medicine All subscribers to the consumer magazine All subscribers who responded to the survey This is the correct answer. Voluntary response sampling was used. no Those who are not subscribers to the consumer magazine I and II
By some estimates, about 20% of all males have a particular defect related to their eyes. How would you assign random numbers to conduct a simulation based on this statistic?
Use random digits 0-99; let 00-19=defect, 20-99=no defect
You're pretty sure that your candidate for class president has about 75% of the votes in the entire school. But you're worried that only 100 students will show up to vote. How often will the underdog (the one with 25% support) win? To find out, you set up a simulation. a) Describe how you will simulate a component. b) Describe how you will simulate a trial. c) Describe the response variable.
Use two random digits, giving 00-74 a vote for your candidate and 75-99 for the underdog. Examine 100 two-digit random numbers, and count how many people voted for each candidate. Whoever gets the majority of votes wins that trial. The response variable is whether the underdog wins or not.
Sammy's Salsa, a small local company, produces 160 cases of salsa per day. Each case contains 10 jars and is imprinted with a code indicating the date and batch number. To help maintain consistency, at the end of each day, Sammy selects jars of salsa, weighs the contents, and tastes the product. Complete parts a through c below.
Using a computer, generate a random number from 1 to 160. Select that case of salsa. Inspect all jars in that case. Divide the day into a certain number of time periods. Randomly select a case from each time period. Inspect a randomly chosen jar from each case. Weight (quantitative) and taste (categorical)
Government inspectors visit cheese factories unannounced and take samples of milk to test for contamination. If the milk is found to contain dirt, antibiotics, or other foreign matter, the milk will be destroyed and the factory reinspected until purity is restored. Would simple random sampling be appropriate for selecting factories for inspection? If so, explain how it would be done. If not, explain why it is not appropriate.
Yes. Inspectors would have a list of all these factories. They could simply number them and have a random number generator select an appropriate number of factories to inspect.
A friend of yours is a very busy woman. If you call her to study for class in the evening, she does not answer the phone 60% of the time, she answers and says no 20% of the time, and she answers and says yes 20% of the time. You want to create a simulation model for the number of nights in a week the two of you get together to study for class. Simulating a phone call on one day of the week is called ____________.
a component of the simulation.
An amusement park has opened a new roller coaster. It is so popular that people are waiting for up to 3 hours for a 2-minute ride. Concerned about how patrons (who paid a large amount to enter the park and ride on the rides) feel about this, they survey every person in line for the roller coaster. Complete parts a through d below.
a convenience sample all the patrons willing to wait in line for the roller coaster no, the sample is not likely to be representative of everyone in the park the patrons who are not willing to wait in line for the ride
A friend of yours is a very busy woman. If you call her to study for class in the evening, she does not answer the phone 60% of the time, she answers and says no 20% of the time, and she answers and says yes 20% of the time. You create a simulation model for the number of nights in a week the two of you get together to study for class, and run the simulation for 20 trials. A report on your results could include
a dotplot of the number of times per week you studied together over the 20-week simulation.
Coffee stations in offices often just ask users to leave money in a tray to pay for their coffee, but many people cheat. On some days, researchers at Newcastle University replaced the picture of flowers on the wall behind the coffee station with a picture of staring eyes. They found that the average contribution increased significantly above the well-established standard when people felt they were being watched, even though the eyes were patently not real. Complete parts a through c below.
an experiment, controlled the situation by determining which picture was used. picture that is displayed behind the coffee station, average contribution from the participants. The differences in money collected were larger than could be reasonably attributed to usual day-to-day variation.
Question content area top Part 1 Suppose that 30% of the cereal boxes contained a picture of a bear, 20% contained a picture of a horse, and the rest contained a picture of a goat. Suppose that you really want the bear picture. How many boxes of cereal do you need to buy to be pretty sure of getting at least one picture of a bear? Let "pretty sure" mean that you get at least one picture of a bear in 90% of trials. Run a simulation with 10 trials using the accompanying sequences of random numbers.
bear, horse, goat
In a large city school system with 42 elementary schools, the school board is considering the adoption of a new policy that would require elementary students to pass a test in order to be promoted to the next grade. The PTA wants to find out whether parents agree with this plan. Listed below are some of the ideas proposed for gathering data. For parts a through d, indicate what kind of sampling strategy is involved and what (if any) biases might result. Assume the schools are homogeneous and differ from each other.
cluster sampling undercoverage bias could result since the parents in the sample may not be representative of all parents systematic sampling bias could result only if the sampling strategy is not followed as described a voluntary response sample voluntary response bias will result since parents parents who feel strongly about the issue will respond an attempt at a census nonresponse bias could result since not all parents will respond
Some friends of yours in a political science class are angry about a new town ordinance restricting off-campus parties. They make an online survey asking students' opinions. This type of sampling might be classified as a __________ sample.
convenience
The website www.buzzfeed.com is an entertainment and news website, known for fun and silly content. In 2021, a Buzzfeed poll asked visitors of the site to select between 8 different ice cream choices. Of the 9072 respondents, 16.7% said that they wanted vanilla ice cream. Complete parts a through c.
convenience sample. The percentage of all people who prefer vanilla ice cream over the 7 other choices in the poll You should place no confidence in the result of this sample.
An experiment tested the ability of maggots to remove dead tissue from open wounds that would not heal on their own. Sterile maggots were placed in a small pouch which, in turn, was placed on the wound. 75 men with wounds on their lower limbs were randomly assigned to receive either a traditional surgical treatment or maggot therapy. After eight days, the percentage of dead tissue in the wounds that underwent maggot treatment was 53.5%, compared to 67.8% with the surgical treatment. Neither patients nor the doctors evaluating the wounds knew which therapy had been applied. (Patients were blindfolded as bandages were changed.) Surprisingly, the number of patients that reported a crawling sensation in their wound was about the same in both groups! Complete parts a through d below.
double blind, experiment, randomized Yes, such an experiment is an appropriate way to determine whether maggots were as effective as surgery. The 75 men with wounds on their lower limbs Traditional surgical procedure, Sterile maggots Percentage of dead tissue
Subjects in an experiment for an herbal cold medicine either receive a tea made from the herb, or a placebo tea. Their doctor also doesn't know which tea was distributed to which subject. This experiment is ____________.
double blinded.
At its website, the Gallup Poll publishes results of a new survey each day. Scroll down to the end of the published results and you'll find a statement that includes words such as those shown below. Results are based on telephone interviews with 1,008 national adults, aged 18 and older, conducted on April 2-5, 2007 ... In addition to sampling error, question wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls. Complete parts (a) and (b).
everyone in the nation that is 18+ years old everyone in the nation that is 18+ years old with a telephone
Over a 5-month period, among 82 people with a circulatory disorder, patients who were given a high dose (723 mg/day) of a certain dietary supplement improved more than those given a placebo. The researchers kept track of and were aware of the dosages of supplements and placebos. Complete parts a through h.
experiment The subjects are 82 people with a circulatory disorder.
Having at least one 30-minute warm-up session per week resulted in a drastic reduction in tears in the quadriceps tendon. In a study involving about 3000 adolescent girls' baseball players in a certain country, one group was randomly assigned to warm up with a neuromuscular exercise session. This group had 72% fewer quadriceps tendon tears than the control group. Complete parts a through d below.
experiment random girls were assigned the specific exercise. Having at least one 30-minute neuromuscular warm-up exercise session per week and not having a warm-up session Number of quadriceps tendon tears in each group Less variation in quadriceps tendon tears would be expected among players of the same gender in the same sport. The conclusions of the study cannot be generalized to boys, girls in other countries, or to players of other sports.
A variable whose levels are controlled by the experimenter is called a ______________.
factor
Occasionally, when Josh fills his car with gas, he figures out how many miles per gallon his car got. He wrote down those results after eight fill-ups in the past few months. Overall, it appears that his car gets 28.6 miles per gallon. a) What statistic has he calculated? b) What is the parameter he is trying to estimate? c) How might his results be biased? d) When the Environmental Protection Agency (EPA) checks a car like his to predict its fuel economy, what parameter is it trying to estimate?
mean, the last eight fill-ups of his car mean, his car recent driving conditions may not be typical. mean, all cars that make and model
A student has trouble getting to work on time. She figures that she is late 30% of the time, on time another 60% of the time, and early only 10% of the time. Since there are three cases, she proposes the following simulation model. Randomly generate numbers 1, 2, or 3, with a 1 representing being "late," a 2 representing being "on time," and a 3 representing being "early." Is this an appropriate model for this situation?
no
A student who likes to gamble wants to build a simulation model for playing roulette. The student sets up the simulation assuming that the bet is on red. This means the simulation should show a "win" with probability 18/38. The student proposes to generate random numbers that are either 1 or 2, with a 1 representing a "win" and a 2 representing a "loss." Is this an appropriate model for this situation?
no
before flipping a coin, your friend ask you to "call it" is your choice (heads or tails random)
no, probably not random many people tend to always pick one side over the other
A friend asks you to quickly name a number between 1 and 10.Is the number random?
no, since most people have a bias towards certain numbers
Use the report of statistical research to identify the following. a) whether it was an observational study or an experiment b) whether it was retrospective or prospective c) the subjects studied, and how they were selected d) the parameter of interest e) the nature and scope of the conclusion regarding what the results may or may not indicate Researchers studied 90 men and 53 women with major depression, and 75 men and 95 women with no depression. They showed the subjects a sad film, noting whether or not the subjects cried. Men cried more often than women, but there were no significant differences between the depressed and nondepressed groups.
observational study prospective men and women with and without major depression; unknown selection process. Frequency of crying in response to a sad movie There is no apparent difference in crying response to sad movies for the depressed and the nondepressed groups.
Use the report of statistical research to identify the following. a) whether it was an observational study or an experiment b) whether it was retrospective or prospective c) the subjects studied, and how they were selected d) the parameter of interest e) the nature and scope of the conclusion regarding what the results may or may not indicate In a test of 500 men, those with high blood pressure did worse on tests of memory and reaction time than those with normal blood pressure.
observational study prospective men with high and normal blood pressure; unknown selection process. Memory and reaction time There is no way to know that high blood pressure caused subjects to do worse on memory and reaction time tests.
Researchers have linked an increase in the incidence of stomach cancer in Greece to a pollutant released by an industrial accident. The study identified 963 people who lived near the site of the accident and were under age 40 at the time. Fourteen of the people had developed stomach cancer at an unusually young average age of 45. Medical records showed that they had heightened concentrations of the pollutant in their blood and that each tenfold increase in the level of the pollutant was associated with a doubling of the risk of stomach cancer. Complete parts a through e below.
observational study retrospective People exposed to the pollutant from the industrial accident The risk of stomach cancer As there is no random assignment, there is no way to know that the levels of the pollutant caused the increase in stomach cancer.
In 1986 a popular science magazine reported that a study of women in Australia indicated that having sons shortened the lifespans of mothers by about 25 weeks per son, but that daughters helped to lengthen the mothers' lives. The data came from church records from the period 1676 to 1832. Answer parts a) through e) below.
observational study retrospective women in Australia The selection process is unknown The parameter of interest is difference in average lifespan between mothers of sons and mothers of daughters. For this group, having sons was associated with a decrease in lifespan of an average of 25 weeks per son, while having daughters was associated with an unspecified increase in lifespan. Without random assignment, there is no way to know that having sons caused a decrease in lifespan.
A study based on data in which no one manipulates any experimental factors is called an _______________.
observational study.
In countries known to contain lower levels of vitamin D, there were more reported cases of a particular type of rare disease. Some researchers hypothesized that a deficiency in vitamin D would increase the risk of developing the disease. These researchers compared vitamin D levels in blood samples from 170 military personnel who have developed the disease with blood samples of nearly 320 who have not. The samples were taken, on average, five years before the disease was diagnosed. Those with the highest blood vitamin D levels had a 56% lower risk of developing the disease than those with the lowest levels. Complete parts a) through d).
observational, retrospective Yes, because the disease in the study is a rare disease, a retrospective study is appropriate. military personnel The level of vitamin D for the subject and whether or not the subject developed the disease
A "fake" treatment that looks just like the treatment being tested is called a _____
placebo
You are doing a study for a non-profit group helping at-risk children in your city. Suppose you know that 14.2% of the children in your city live in poverty. This percentage is an example of a __________.
population parameter.
A friend of yours is a very busy woman. If you call her to study for class in the evening, she does not answer the phone 60% of the time, she answers and says no 20% of the time, and she answers and says yes 20% of the time. You want to create a simulation model for the number of nights in a week the two of you get together to study for class. In this example, the response variable is
quantitative- the number of nights you study over the course of 7 nights
You are trying to study the amount of financial aid students at your University receive. You sample 50 students and find out the average size of their financial aid packages. The average of your sample is a __________.
sample statistic.
A friend of yours is a very busy woman. If you call her to study for class in the evening, she does not answer the phone 60% of the time, she answers and says no 20% of the time, and she answers and says yes 20% of the time. You want to create a simulation model for the number of nights in a week the two of you get together to study for class. In this example, a trial consists of running the simulation for ____________.
seven nights, or seven times.
When you sample so that every combination of individuals in your population has an equal chance of being chosen you are taking a __________.
simple random sample.
A friend of yours in your intro stats class obtains permission to randomly sample the University student body to conduct a satisfaction survey on some recent changes to the enrollment process. She randomly samples 50 first-year students, 50 first-year students, 50 second-year students, 50 third-year students, and 50 fourth-year students. This is an example of a
stratified
In a large city school system with 34 elementary schools, the school board is considering the adoption of a new policy that would require elementary students to pass a test in order to be promoted to the next grade. The PTA wants to find out whether parents agree with this plan. Listed below are some of the ideas proposed for gathering data. For parts a through d, indicate what kind of sampling strategy is involved and what (if any) biases might result. Assume the schools are homogeneous and differ from each other.
stratified sampling bias could result only if the sampling strategy is not followed as described a simple random sample bias could result only if the sampling strategy is not followed as described systematic sampling this is probably a reasonable design convenience sampling undercoverage bias could result since only parents involved with the pta will be sampled
The Environmental Protection Agency took a map of a region near a former industrial waste dump and placed a grid of 525 squares on it. They collected soil samples from every 16th square and checked each for evidence of toxic chemicals. Complete parts a) through c) below.
systematic sample There is no bias associated with this procedure. Use a stratified random sample, stratifying by distance from the dump site.
How long is your arm compared with your hand size? Put your right thumb at your left shoulder bone, stretch your hand open wide, and extend your hand down your arm. Put your thumb at the place where your little finger is, and extend down the arm again. Repeat this a third time. Now your little finger will probably have reached the back of your left hand. If your arm is less than four hand widths, turn your hand sideways and count finger widths until you reach the end of your middle finger. a) Suppose you repeat your measurement 15 times and average your results. What parameter would this average estimate? What is the population? b) Suppose you now collect arm lengths measured in this way from 8 friends and average these 9 measurements. What is the population now? What parameter would this average estimate? c) Do you think these 9 arm lengths are likely to be representative of the population of arm lengths in your community? In the country? Why or why not?
the length of your arm, all possible measurements of your arm. all possible measurements of your arm and your friends' arms, mean, all possible measurements. The sample would not be representative of either the community or the country because a group of friends might not be as diverse as the population.
Use random numbers from 1 through 13 to represent the denominations of the cards in a five-card poker hand.
the likelihood of, for example the first ace in the hand is not the same as for the second or third or fourth. but with the simulation the likelihood is the same for each
Use a random integer from 0 to 5 to represent the number of boys in a family of 5 children.
the number of boys in a family of 5 children are not equally likely, for example having 5 boys is less likely than having 3 boys. the simulation assigns the same likelihood to each event
For the following report about a statistical study, identify (if possible) a) the population; b) the population parameter of interest; c) the sampling frame; d) the sample; e) the sampling method, including whether or not randomization was employed; f) who (if anyone) was left out of the study; and g) any potential sources of bias you can detect and any problems you see in generalizing to the population of interest. Researchers waited outside a bar they had randomly selected from a list of such establishments. They rolled a twenty-sided die and it came up 3, so they stopped the third adult who came out of the bar, then every 16th adult after that, and asked whether they thought drinking and driving was a serious problem.
the population of all adults who would visit a bar The population parameter is the proportion of adults who think drinking and driving is a serious problem. The sampling frame is patrons from the chosen bar The sample is every 16th adult leaving the bar. The sample method used was systematic sampling the sampling method used is random any adults who would not go to a bar undercoverage and response bias adults never leaving a bar may think drinking and driving is less of a problem than other adults
A recent public survey asked the following question. "Many people believe this playground is too small and in need of repair. Do you think the playground should be repaired and expanded even if that means raising the entrance fee to the park?" Describe two ways this question may lead to response bias.
the question mentions higher fees, which could make people reject improvements to the playground. The statement points out problems the respondent may not have noticed, and might lead them to feel they should agree.
A group of students in your intro stats class design an experiment to test whether popcorn stored in the freezer pops better (fewer kernels left un-popped) than room temperature popcorn. This group numbers 20 bags of popcorn, and uses a random number generator to select 10 random numbers between 1 and 20. The selected bags are put in the freezer treatment group. This is an example of a completely randomized design? True or False.
true
You recently began an internship at your local chapter of savethepigeons.com. Concerned about a city ballot initiative dealing with the environment, you conduct a telephone survey of local residents. What are some possible sources of bias in your results?
undercoverage of the population response bias non-response bias
Anytime a survey is conducted, care must be taken to avoid undercoverage. Suppose a firm selects 500 names from a city phone book, calls their homes between noon and 4 p.m., and interviews whoever answers, anticipating contacts with at least 200 people. Complete parts (a) through (e).
undercoverage, sampling frame The phone numbers could be randomly generated, and the calls could be at random times of the day. That way people with unlisted numbers and people away from their homes from noon to 4 p.m. could be included. included, included, excluded Follow-up of this type greatly improves the chance that a selected household is included, increasing the reliability of the survey. Random dialers allow people with unlisted phone numbers to be selected. Time of day will still be an issue, as will people without phones.
A company employs 14 project managers, 375 inspectors, and 367 assemblers. In an effort to keep informed about any possible sources of employee discontent, management wants to conduct job satisfaction interviews with a simple random sample of employees every month. Complete parts a through e below.
undercoverage,project managers. Assign a random number to each employee and use a list of random numbers or software to choose the employees for the sample. The responses for each group may be different, so it is important that the data for each group can be examined individually. The company should use stratified sampling and randomly choose a proportionate number of people from each group. pagilarulo, thomas
If you create an online survey, individuals can choose on their own whether to participate in the sample. This causes a form of bias called __________.
voluntary response.
A student has trouble getting to work on time. She figures that she is late 30% of the time, on time another 60% of the time, and early only 10% of the time. She proposes the following simulation model. Randomly generate integers between 0 and 9 with the digits 0-2 representing being "late," the digits 3-8 representing being "on time," and the digit 9 representing being "early." Is this an appropriate model for this situation?
yes
names are selected out of a hat to decide roommates in a dormitory.Is your roommate for the year random?
yes, the new roommate cannot be predicted before names are draen
Flip a coin to decide who takes out the trash. Is who takes out the trash random?
yes, who takes out the trash cannot be predicted before the flip of the coin
In many state lotteries you can choose which numbers to play. Consider a common form in which you choose 5 numbers. Will the strategy of generating random numbers using a computer or calculator and playing those improve your chance of winning? Explain.
No, because each number drawn is equally likely and independent of the others, so this set of numbers is just as likely as any other in the next drawing.
