Comparing Different Studies

Transcription

1 32 LESSON Comparing Different Studies Types of Studies UNDERSTAND By studying a small group within a larger group, you can make inferences about the larger group. The larger group, called the population, is a collection of people or things that have something in common. All registered voters in Atlanta and all lightbulbs produced in a factory yesterday are examples of populations. The smaller group taken from the population is called the sample. The best way for you to ensure that a sample is representative of the population is to gather members of the sample at random. A parameter is a measureable characteristic of a population. For example, if a senior class contains 297 boys and 363 girls, saying that 45% of the senior class is boys is stating a parameter because it describes the entire class. Parameters can be difficult to obtain, particularly for large populations. So, scientists and mathematicians often try to gather statistics. A statistic is a measureable characteristic of a sample. For example, imagine that senior boys in a random sample have an average height of 5 feet 10 inches. That is a statistic because it applies only to the sample. UNDERSTAND Statistical studies can be done in different ways. In an experimental study, the researcher designs and performs an experiment and then observes the responses to the treatment administered. In a controlled experiment, the researcher will divide the sample into a control group and one or more treatment groups. The control group does not receive any manipulation, while the treatment groups do. Imagine that a drug company wants to test the efficacy of a new cholesterol drug. The control group is given a placebo instead of the drug, whereas the treatment group gets 40 milligrams per day of the drug. In an experiment, a response variable measures an outcome. In the cholesterol study, the response variable is the patients cholesterol levels. It is a quantitative (numerical) variable; you can think of it as the dependent variable. An explanatory variable is an influencing factor. Whether or not patients received the cholesterol drug is an explanatory variable. This can be a categorical variable or a numerical variable; you can think of it as the independent variable. Though it is not always possible, researchers will try to have only one explanatory variable in a controlled experiment. UNDERSTAND In an observational study, the researcher makes observations about a sample without administering any treatment. For example, a researcher is interested in knowing whether job applicants who smile more are more successful in interviews. In an observational study, the researcher could sit in on a round of first interviews and record the number of times each candidate smiles. Then, she could compare her observations to the list of candidates invited back for a second interview. A survey is a type of observational study used to collect and analyze information. It may involve giving out questionnaires or conducting interviews. 318 Unit 7: Inferences and Conclusions from Data

2 Connect For each situation, determine the population, sample, and type of study. If appropriate, identify a parameter or statistic. 1 A company wants to test its new video game. The company randomly selects 50 teenagers from a local high school to play the game while researchers watch and record the teenagers reactions. The population is any potential purchaser of the game. The sample is the 50 teenagers selected at random. This is an example of an observational study. 2 A city parks department is deciding what to do with a vacant lot. Researchers send out a questionnaire to all of the homes within one mile of the lot, asking about what types of projects the residents would prefer. 3 A grocery store owner is considering whether to purchase a sound system for the store. He wants to determine whether playing music in the store has any effect on shoppers. On Monday, he plays music in the store. Sales on that day are 9% higher than on the previous Monday. On Tuesday, he does not play music. Sales on that day are 1% lower than on the previous Tuesday. The population is people who shop at the store. The sample is the customers who shopped on that Monday and Tuesday. The statistics are that sales increased by 9% when music was played and declined by 1% when there was no music. This is an example of an experimental study. DISCUSS The population is the residents within one mile of the lot. A parameter in this situation is that the population lives within one mile of the lot. The sample is those residents who choose to complete and send in the questionnaire. This is an example of a survey. In the experimental study, what is the treatment? Who could be considered the control group? Lesson 32: Comparing Different Studies 319

3 Sampling and Bias UNDERSTAND In statistics, bias is present when the results of a study have been influenced in some manner. Bias can happen as a result either of the sampling method or of the manner in which an experiment is performed. There are various ways to choose a sample from a population. In a random sample, each member of the population has an equal chance of being chosen. You might create a random sample by selecting names at random and without looking from pages throughout a phone book. A systematic sample uses some sort of system to choose from the population. You might use systematic sampling by choosing every 20th name from the phone book. A stratified sample separates the population into groups based on a characteristic and then chooses randomly from within each group. You could create a stratified sample by choosing 50 students at random from each grade at your school. Each of these methods produces a sample that is random and, therefore, unbiased. Some sampling methods are not random and may lead to bias. A convenience sample is made up of subjects who are convenient to study. If you only surveyed your family and your closest neighbors, you would be using a convenience sample. Your sample is drawn from a population, but it may not be representative of that population. A voluntary sample is made up of people who volunteer to participate. An example of a voluntary sample is voters for a TV talent show. Voluntary samples often draw only people with strong opinions on the subject and so may not be representative of the population. UNDERSTAND Another common way in which bias occurs is as a result of how questions are asked in an experiment or survey. Consider the following examples, which ask a survey question in different ways. Would you agree that students should carpool to school? Would students carpooling to school save money and protect the environment? Do you think students should carpool to school? In the first question, the phrase would you agree leads the respondent toward agreeing with the statement. In the second question, the respondent is given specific, positive things to think about saving money and protecting the environment that may influence his or her answer. The third question simply asks for the respondent s opinion on the matter. It is a neutral and unbiased question. 320 Unit 7: Inferences and Conclusions from Data

4 Connect A school is considering whether to build a football stadium or to start an orchestra program. Members of the student council placed questionnaires in the gym locker rooms and asked students to complete and return the questionnaires. The results showed that 88% of students favor building the football stadium. What type of sampling was done? Is it a random sample or a biased sample? How could the study be improved? 1 Identify the type of sampling used. The student council members only distributed questionnaires once and in just a few places. You can call this a convenience sample. It is also a voluntary sample since the questionnaires were returned only by students who chose to do so. 2 Is the sample random or biased? Both of the sampling methods used convenience and voluntary lead to biased samples. The questionnaires were in gym locker rooms, so the students who found them are much more likely to be interested in athletics than the average student. Thus, those students probably would be more inclined toward a football stadium than toward an orchestra program. 3 Determine how the study could be improved. Any kind of random sampling would make the results of the study more representative of the entire student body. For example, creating a stratified sample and interviewing the members of the sample would yield much better results. DISCUSS The sample is biased in favor of the football stadium. Write a biased question that might appear on the survey. Lesson 32: Comparing Different Studies 321

5 EXAMPLE A Runners who suffered hamstring injuries were randomly assigned to one of two physical therapy programs. Program 1 incorporates a new type of heat therapy in addition to traditional therapy. Program 2 is the existing program of traditional therapy. Runners who participated in program 1 returned to running in a mean of 33.4 days. Those assigned to program 2 returned to running in a mean of 22.2 days. Identify the population (the subjects studied), the control and treatment groups, the explanatory variable, and the response variable. Draw a conclusion from the study. 1 Identify the population, control group, and treatment group. The population is the runners who had hamstring injuries. The control group is the runners assigned to program 2, who received the traditional course of therapy. The treatment group is the runners assigned to program 1, who received the additional, new heat therapy. 2 Identify the explanatory and response variables. 3 Draw a conclusion from the study. The explanatory variable is the presence or absence of heat therapy in a program. The response variable is the number of days it took for each runner to get back to running. The runners in program 2 returned to running an average of 11.2 days sooner than those in program 1. The new heat therapy does not seem to help injured runners. In fact, it appears to slow down the recovery process considerably and probably should not be used. DISCUSS Was the study biased in any way? 322 Unit 7: Inferences and Conclusions from Data

6 EXAMPLE B A pharmacy chain wants to determine whether shoppers are more likely to buy sunscreen if it is located near the checkout line. The experimental group consists of 10 stores in Illinois in which the sunscreen was moved next to the checkout line. The control group consists of 10 stores in Florida in which the sunscreen was not moved. After 1 month, the Florida stores sold 21% more sunscreen than the Illinois stores. The chain concluded that moving the sunscreen closer to the checkout line does not increase sales. Identify any flaws in the experiment design, and describe how they could be corrected. 1 Identify flaws in the experiment. All stores in the control group are in Florida, and all stores in the treatment group are in Illinois. Florida is generally warmer and sunnier than Illinois; Florida also contains many more miles of beach than Illinois. It is highly likely that people in Florida buy more sunscreen than people in Illinois, no matter where the sunscreen is located in a store. The experiment is flawed because it compares sunscreen sales across stores in very different climates. 2 Describe how the flaw could be corrected. DISCUSS To eliminate the problem of different environments, the pharmacy chain decided to test its theory in just one store. In August, the sunscreen was moved to be near the checkout line. In September, the sunscreen was moved back to its original location. Is this experiment flawed? In what way? The pharmacy chain would get much better experimental results by setting both the control group and the treatment group in either Florida or Illinois. That way, the sales comparison would involve stores in similar climates. Lesson 32: Comparing Different Studies 323

7 EXAMPLE C A local city councilwoman sent the following survey to her constituents. The crime rate in our city is down 10% as compared to 5 years ago, and the rate of obesity is rising. Answer yes or no to the following questions. Do you think the bloated police budget should be cut to pay for new tennis courts that will be free for residents? Do you think tennis courts are a great way to encourage exercise in the community? Do you think we have more police than we need? Identify any flaws in the survey, and describe how they could be corrected. 1 Identify flaws in the survey. 2 Examine the questions. One flaw is in the sampling method. By sending out the survey, the councilwoman created a self-selected (or voluntary) sample the only responses she would receive are from people who chose to complete the survey. Another flaw is that the survey presents leading information before the survey questions. By stating that crime has dropped and obesity has increased, the survey may influence participants to respond in a particular way. The wording of each question creates bias in the survey. In the first question, the word bloated is subjective and implies that the current police force is wasteful and too big. Also, by saying that tennis courts will be free for residents, the question makes it more likely that respondents will see the courts as a benefit to themselves. In the second question, the word great may lead constituents to agree with the statement. 3 Describe how the survey s flaws could be corrected. A better sampling method would be going to randomly chosen houses to conduct interviews. No information about crime and obesity rates should be given with the survey. Finally, the survey questions should be rewritten in an unbiased manner. DISCUSS The third question plants the idea that the city has more police than necessary. The aggregate effect of the questions is to lead constituents to think that the city needs fewer police since crime rates have fallen. But it is certainly possible that crime rates have fallen because there is a greater police presence on the streets. What are some possible reasons that the councilwoman would send out a biased survey? 324 Unit 7: Inferences and Conclusions from Data

8 READ Problem Solving A researcher is interested in knowing whether black dogs are more or less likely to be adopted from animal shelters than dogs of other colors. He gathered data for the previous month from three randomly selected shelters. The data are displayed in the table below. Black Other Colors Total Dogs at Shelters Adopted Dogs Total Identify the population, sample, and type of study. Complete the table by finding the missing totals. Calculate statistics and draw a conclusion. Look for bias in the experiment. PLAN Find the percent of dogs in the sample that are black. Then, compare that to the percent of adopted dogs that are black. SOLVE The population in the study is. The sample in the study is. This is an To the nearest tenth, about To the nearest tenth, about study. % of dogs in the sample were black. % of adopted dogs were black. Based on the data, black dogs are likely to be adopted as dogs of other colors. Do you detect any bias in the experiment? Explain. CHECK Find each ratio. Write each answer as a decimal to the nearest hundredth. The ratio of black dogs to dogs of other colors in the sample is about. The ratio of adopted black dogs to adopted dogs of other colors is about. Black dogs are to be adopted as dogs of other colors. Lesson 32: Comparing Different Studies 325

9 Practice Write experimental, observational, or survey to describe each study type. Then, identify each aspect of the study, and answer any questions. 1. A math teacher gave a quiz to every third student who walked into her class. The quiz contained five questions, and the average score was 3.7 out of 5. study type: population: sample: sampling method: Is 3.7 out of 5 a parameter or a statistic? REMEMBER A parameter describes a population, and a statistic describes a sample. 2. A baseball coach divided his team randomly into two groups. One group practiced with regular bats, while the other group practiced with weighted bats. The group using weighted bats hit the ball an average of 2.3 meters farther than the group using regular bats. study type: population: control group: treatment group: Is 2.3 m farther a parameter or a statistic? 3. Ginny wants to know what students in her school think about the new cafeteria foods. She interviewed 10 students from her English class. study type: population: sample: sampling method: HINT How difficult was it for Ginny to survey students from her English class? 326 Unit 7: Inferences and Conclusions from Data

10 Choose the best answer. 4. Which question is the least biased? A. Do you think 10-year-old boys should be allowed to play football even if they might sustain serious head injuries? B. Do you agree that karate is too dangerous for 5 year old boys? C. Do you think that 15 year olds should get jobs instead of being lazy? D. Do you think 12 year olds should be allowed to babysit younger siblings? 5. Which sampling technique is most likely to produce a random, representative sample? A. choosing everyone in the drama club B. choosing every 10th person in the phone book C. choosing the first 10 people you see D. asking for volunteers Solve. 6. A farmer has decided to try a new fertilizer on part of her land. As the control group, she uses her current fertilizer on the potato field. As the treatment group, she uses the new fertilizer on the lettuce field. At the end of the harvest, she had 570 lb of potatoes but only 293 lb of lettuce. She concluded that the new fertilizer is less effective than her current brand. Identify any flaws in the study, and describe how they could be corrected. Lesson 32: Comparing Different Studies 327

11 Identify the subjects, response variable, and explanatory variable. Draw a conclusion, and include all explanatory variables in your conclusion. 7. Scientists wanted to determine whether a daily dose of vitamin C can prevent a cold. They randomly assigned 100 parents to group A and 100 to group B. The parents in group A took 90 mg of vitamin C each day for a month, and the parents in group B took no vitamin C. All parents reported whether they got a cold during the month and whether someone in their house had a cold during the month. The results are shown. Developed a Cold Household Member with Cold Group A Group B subjects: response variable: explanatory variable(s): conclusion: 8. To test the effectiveness of a diet cat food, scientists divided a group of overweight cats into a control and treatment group. The cats in the control group ate their regular food. The cats in the treatment group received a measured portion of diet cat food twice a day. After two months, the cats in the treatment group had lost an average of 1.2 lb, and the cats in the control group had gained an average of 0.1 lb. subjects: response variable: explanatory variable(s): conclusion: 328 Unit 7: Inferences and Conclusions from Data

12 Solve. 9. The city council is deciding between building a new elementary school to ease overcrowding or building a new community pool. The council members went to a local playground, conducted interviews with willing participants, and found that 73% of respondents favored the new school over the pool. Identify any flaws in the study, and describe how they could be corrected. 10. CRITIQUE A track coach wants to know whether wearing a new type of lighter shoe makes sprinters run faster. He times all members of the track team in a 100-yard dash. Then, he has them all switch to the lighter shoe and times them again. He finds that, on average, the runners completed the run 0.03 sec faster while wearing the lighter shoe. Nadia wrote: This was an observational study, because the coach was observing the runners times. Is Nadia correct? Explain. 11. DESIGN A school district wants to conduct a survey to determine the number of high school juniors in the district who are planning to attend college after graduation. State the objective of the survey. Suggest a population. Write two unbiased survey questions. Lesson 32: Comparing Different Studies 329