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# Chapter 1 notes Mgt 213

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This 21 page Class Notes was uploaded by Stephanie knubis on Thursday July 21, 2016. The Class Notes belongs to Mgt 213 at Marian University taught by Andrew books in Summer 2016. Since its upload, it has received 179 views. For similar materials see Principles of Management in Business at Marian University.

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MAT 123: Fundamentals of Statistics Student Note Sheets Part 1: Getting the Information You Need Chapter 1: Data Collection Section 1.1: Introduction to the Practice of Statistics Learning Objective 1. Define Statistics and statistical thinking. Definitions Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. In addition, statistics is about providing a measure of confidence in any conclusions. Data is a fact or proposition used to draw a conclusion or make a decision. Learning Objective 2. Explain the process of statistics. Definitions The entire group to be studied is called the population. An individual is a person or object that is a member of the population being studied. A sample is a subset of the population that is being studied. A statistic is a numerical summary of a sample. Descriptive statistics consist of organizing and summarizing data. Descriptive statistics describe data through numerical summaries, tables, and graphs. Inferential statistics uses methods that take a result from a sample, extend it to the population, and measure the reliability of the result. A parameter is a numerical summary of a population. Example 1: Determine whether the underlined value is a parameter or a statistic. 1. Calculus Exam: The average score for a class of 28 students taking a calculus midterm exam was 72%. 2. Drug Use: In a national survey on substance abuse, 10.0% of respondents aged 12 to 17 reported using illicit drugs within the past month. 3. Moonwalkers: Only 12 men have walked on the moon. The average age of these men at the time of their moonwalks was 39 years, 11 month, 15 days. 4. Public Knowledge: Interviews of 100 adults 18 years of age or older, conducted nationwide, found that 44% could state the minimum age required for the office of U.S. president. The Process of Statistics 1. Identify the research objective. A researcher must determine the 1 questions(s) he or she wants answered. The question(s) must be detailed so it identifies the population that is to be studied. 2. Collect the data needed to answer the question(s) in (1). Conducting research on an entire population is often difficult and expensive, so we typically look at a sample. The step is vital to the statistical process, because if the data are not collected correctly, the conclusions drawn are meaningless. Do not overlook the importance of appropriate data collection. 3. Describe the data. Descriptive statistics allow the researcher to obtain an overview of the data and can help determining the type of statistical methods the researcher should use. 4. Perform inference. Apply the appropriate techniques to extend the results obtained from the sample to the population and report a level of reliability of the results. Example 2: Using the Process of Statistics, answer the following questions. Retirement Planning: The Principal Financial Group conducted a survey of 1172 employees in the United States between July 28, 2010 and August 8, 2010. They asked if the employees were currently participating in the employer-sponsored automatic payroll deduction for a 401(k) plan to save for retirement. Of the 1172 employees surveyed, 27% indicated they were participation. The Principal Group reported that 27% of all employees in the United States participate in automatic payroll deduction for a 401(k) plan to save for retirement with a 4% margin of error and 95% confidence. a. What is the research objective? b. What is the population? c. What is the sample? d. List the descriptive statistics. e. What can be inferred from the survey? Learning Objective 3. Distinguish between qualitative and quantitative variables. Definitions Variables are the characteristics of the individuals within the population. Qualitative, or categorical, variables allow for classification of individuals based on some attribute or characteristic. 2 Quantitative variables provide numerical measures of individuals. The values of a quantitative variable can be added or subtracted and provide meaningful results. Example 3: Classify the variable as qualitative or quantitative. 1. Number of siblings. 2. Number on a football player’s jersey. 3. Assessed value of a house. 4. Student ID number. Learning Objective 4. Distinguish between discrete and continuous variables. Definitions A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. The term countable means that the values result from counting, such as 0, 1, 2, 3 and so on. A discrete variable cannot take on every possible value between any two possible values. A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable. A continuous variable may take on every possible value between any two values. Qualitative Quantitative Variables variables Discrete Continuous Variables Variables Example 4: Determine whether the quantitative variable is discrete or continuous. 1. The volume of water lost each day through a leaky faucet. 2. Number of Sequoia trees in a randomly selected acre of Yosemite National Park. 3 3. Internet connection speed in kilobytes per second. 4. Air pressure in pounds per square inch in an automobile tire. Definitions The list of observed values for a variable is data. Qualitative data are observations corresponding to a qualitative variable. Quantitative data are observations corresponding to a quantitative variable. Discrete data are observations corresponding to a discrete variable. Continuous data are observations corresponding to a continuous variable. Example 5: Identify the individuals, variables, and data corresponding to the variables. Determine whether each variable is qualitative, continuous, or discrete. BMW Cars. The following information relates to the 2011 model year product line of BMW automobiles. Model Body Style Weight (lb) Number of Seats 3 Series Coupe 3362 4 5 Series Sedan 4056 5 6 Series Convertible 4277 4 7 Series Sedan 4564 5 X3 Sport Utility 4012 5 Z4 Roadster Coupe 3505 2 What are the individuals? What are the variables? What is the data for Body style? What is the data for weight? What is the data for the number of seats? Which variables are qualitative? Which variables are quantitative? Which variables are continuous? Which variables are discrete? 4 Part 1: Getting the Information You Need Chapter 1: Data Collection Section 1.2: Observational Studies Versus Designed Experiments Learning Objective 1. Distinguish between an observational study and an experiment. Definitions An Explanatory Variable is one that explains changes in the response variable, it can be any factor that might influence the response variable. An explanatory variable is another term for an independent variable, but with a subtle difference. An independent variable is completely and totally independent but when it is not known for certain if it is independent, then it is an explanatory Variable. A Response Variable is the focus of the question in a study or experiment. An Observational study measures the value of the response variable without 5 attempting to influence the value of either the response or explanatory variables. That is, in an observational study, the researcher observes the behavior of the individuals without trying to influence the outcome of the study. If a researcher assigns the individuals in a study to a certain group, intentionally changes the value of an explanatory variable, and then records the value of the response variable for each group, the study is a Designed Experiment. Example 1: Determine whether the study depicts an observational study or an experiment. 1. Rats with cancer are divided into two groups. One group receives 5 milligrams (mg) of a medication that is thought to fight cancer, and the other receives 10mg. After 2 years, the spread of cancer is measured. 2. A poll is conducted in which 500 people are asked whom they plan to vote for in the upcoming election. 3. While shopping, 200 people are asked to perform a taste test in which they drink from two randomly placed, unmarked cups. They are then asked which drink they prefer. 4. Conservation agents netted 250 large-mouth bass in a lake and determined how many were carrying parasites. Definitions Confounding in a study occurs when the effects of two or more explanatory variables are not separated. Therefore, any relation that may exist between an explanatory variable and the response variable may be due to some other variable or variables not accounted for in the study. A lurking variable is an explanatory variable that was not considered in a study but that affects the value of the response variable in the study. In addition, lurking variables are typically related to explanatory variables considered in the study. Observational studies do not allow a researcher to claim causation, only association. 6 Example 2: Answer the following questions. Daily Coffee Consumption: Researchers wanted to determine if there was an association between daily coffee consumption and the occurrence of skin cancer. The researchers looked at 93,676 women enrolled in the Women’s Health Initiative Observational Study and asked them to report their coffee-drinking habits. The researchers also determined which of the women had non-melanoma skin cancer. After their analysis, the researchers concluded that consumption of six or more cups of caffeinated coffee per day was associated with a reduction in non-melanoma skin cancer. a. What type of observational study was this? Explain. b. What is the response variable in the study? c. What is the explanatory variable in the study? d. In their report, the researchers stated that “After adjusting for various demographic and lifestyle variables, daily consumption of six or more cups was associated with a 30% reduced prevalence of non-melanoma skin cancer.” Why was it important to adjust for these variables? Learning Objective 2. Explain the Various Types of Observational Studies Three Major Categories of Observation Studies Cross-sectional Studies collect information about individuals at a specific point in time or over a very short period of time. Disadvantage: has limitations. Advantage: cheap and quick to do. Case-control Studies are retrospective, meaning that they require individuals to look back in time or require the researcher to look at existing records. In case- controlled studies, individuals who have a certain characteristic may be matched with those who do not. Disadvantage: requires individuals to recall information from the past and to be truthful. Advantage: relatively quick and inexpensive. Cohort Studies first identifies a group of individuals to participate in the study (the cohort). The cohort is then observed over a long period of time. During this period, characteristics about the individuals are recorded and some individuals will be exposed to certain factors (not intentionally) and others will not. At the end of the study the value of the response variable is recorded for the individuals. Advantage: Most powerful of the observational studies. Disadvantage; require many individuals to participate over long periods of time. Also, many participants drop out over time. 7 Definition A census is a list of all individuals in a population along with certain characteristics of each individual. Part 1: Getting the Information You Need Chapter 1: Data Collection Section 1.3: Simple Random Sampling Learning Objective 1: Obtain a simple random sample. Definitions Random sampling is the process of using chance to select individuals from a population to be included in the sample. A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring. The sample is then called a simple random sample. A frame is a list of all the individuals within the population. In a sample without replacement, an individual who is selected is removed from the population and cannot be chosen again. In a sample with replacement, a selected individual is placed back into the population and could be chosen a second time. For the results of a survey to be reliable, the characteristics of the individuals I the sample must be representative of the characteristics of the individuals in the population. 8 Example 1: As part of a college literature course, students must select three classic works of literature from the provided list and complete critical book reviews for each selected work. Obtain a simple random sample of size 3 from this list. Write a short description of the process you used to generate your sample. Pride and Prejudice Death of a Salesman Scarlet Letter Huckleberry Finn As I Lay Dying The Sun Also Rises The Jungle Crime and Punishment A Tale of Two Cities Example 2: Obtaining a Simple Random Sample Using a Table of Random Numbers. Need Excel spreadsheet, Table 1 and Technology Step-By-Step on page 29. (a) Obtain a simple random sample of size 8 using Table 4 in Appendix A, by hand. (b) Obtain a second simple random sample of size 8 using Table 4 in Appendix A, and a graphing calculator. Step 1: The presidents must be listed (the frame) and numbered from one to 44. Step 2: Eight unique numbers will be randomly selected. The presidents corresponding to the numbers are then selected. Closing your eyes and using your finger, point to a spot on table 1 Appendix 1. Using numbers that are two digit, and less than or equal to 44, pick out the eight corresponding randomly selected presidents. Example 3: Obtaining a Simple Random Sample Using a Technology Step 1: The presidents must be listed (the frame) and numbered from one to 44. Step 2: We randomly select eight numbers using a random number generator. To do this we must first set the seed. The seed is an initial point for the generator to start creating random numbers-like selecting the initial point in the table of random numbers. 9 Part 1: Getting the Information You Need Chapter 1: Data Collection Section 1.4: Other Effective Sampling Methods Learning Objective 1: Obtain a Stratified Sample Definitions A stratified sample is obtained by separating the population into non- overlapping groups called strata and then obtaining a simple random sample from each stratum. The individuals within each stratum should be homogeneous (or similar) in some way. A systematic sample is obtained by selecting every kth individual from the population. The first individual selected corresponds to a random number between 1 and k. A cluster sample is obtained by selecting all individuals within a randomly selected collection or group of individuals. A convenience sample is a sample in which the individuals are easily obtained and not based on randomness. The most popular of the many types of convenience samples are those in which the individuals in the sample are self-selected (they decide themselves to participate in the survey). These are also called voluntary response samples. Example 1: Stratified Sample: The Future Government Club wants to sponsor a panel discussion on the upcoming national election. The club wants to have four of its members lead the panel discussion. To be fair, however, the panel should consist of two Democrats and two Republicans. From the list of current members of the club, obtain a stratified sample of two Democrats and two Republicans to serve on the panel. DEMOCRATS REPUBLICANS 10 Bolden Motola Biouin Ochs Bolt Nolan Cooper Pechtold Carter Opacian DeYoung Redmond Debold Pawlak Engler Rice Fallenbuchel Ramirez Grajewski Salihar Haydra Tate Keating Thompson Khouri Washington May Trudeau Lukens Wright Niemeyer Zenkel To obtain the stratified sample, conduct a simple random sample within each group. Use which ever method you prefer. Learning Objective 2: Obtain a Systematic Sample Steps in Systematic Sampling 1. If possible, approximate the population size, N. 2. Determine the sample size desired, n. N 3. Compute . n 4. Randomly select a number between 1 and k. Call this number p. 5. The sample will consist of the following individuals: p, p + k, p + 2k, ……., p + (n -1)k Example 2: A salesperson obtained a systematic sample of size 20 from a list of 500 clients. To do so, he randomly selected a number from 1 of 25, obtaining the th th number 16. He included in the sample the 16 client on the list and every 25 client thereafter. List the numbers that correspond to the 20 clients selected. Learning Objective 3: Obtain a Cluster Sample Example 3: A quality-control expert wishes to obtain a cluster sample by selecting 10 of 795 clusters. She numbers the clusters from 1 to 795. Using Table I in Appendix A, she closed her eyes and drops a pencil on the table. It points to the digit in row 8, column 38. Using this position as the starting point and proceeding downward, determine the numbers for the 10 clusters selected. The following are a few of the questions that arise in cluster sampling: How do I cluster the population? 11 How many clusters do I sample? How many individuals should be in each cluster? A word of Caution***** Stratified and cluster samples are different. In a stratified sample, we divide the population into two or more homogeneous groups. Then we obtain a simple random sample from each group. In a cluster sample, we divide the population into groups, obtain a simple random sample of some of the groups, and survey all individuals in the selected groups. Convenience samples yield unreliable results because the individuals participating in the survey are not chosen using random sampling. In practice, most large-scale surveys obtain samples using a combination of the techniques just presented. Example 4: identify the type of sampling used. 1. To determine the prevalence of human growth hormone (HGH) use among high school varsity baseball players, the State Athletic Commission randomly selects 50 high schools. All members of the selected high Schools’ varsity baseball teams are tested for HGH. 2. A member of Congress wishes to determine her constituency’s opinion regarding estate taxes. She divides her constituency into three income classes: low-income household, middle-income households, and upper-income households. She then takes a simple random sample of households from each income class. 3. A radio station asks its listeners to call in their opinion regarding the use of U.S. forces in peacekeeping missions. 4. A college official divides the student population into five classes: freshman, sophomore, junior, senior, and graduate student. The official takes a simple random sample from each class and asks the members’ opinions regarding student services. 5. The presider of a guest-lecture series at a university stands outside the auditorium before a lecture begins and hand every fifth person who arrives, beginning with the third, a speaker evaluation survey to be completed and returned at the end of the program. 12 6. 24 Hour Fitness wants to administer a satisfaction survey to its current members. Using its membership roster, the club randomly selects 40 club members and asks them about their level of satisfaction with the club. Part 1: Getting the Information You Need Chapter 1: Data Collection Section 1.5: Bias in Sampling Learning Objective 1: Explain the sources of bias in sampling. Definition If the results of the sample are not representative of the population, then the sample has bias. Non-sampling errors result from under-coverage, nonresponse bias, response bias, or data-entry error. Such errors could also be present in a complete census of the population. Sampling error results from using a sample to estimate information about a population. This type of error occurs because a sample gives incomplete information about a population. Three sources of bias in sampling 1. Sampling Bias means that the technique used to obtain the sample’s individuals tends to factor one part of the population over another. Sampling bias also results due to under-coverage, which occurs when the proportion of one segment of the population is lower in a sample than it is in the population. 2. Nonresponse Bias exists when individuals selected to be in the sample who do not respond to the survey have different opinions from those who do. 3. Response Bias exists when the answers on a survey do not reflect the true feelings of the respondent. Response bias can occur in a number of ways. a. Interviewer Error: b. Misrepresented Answers: c. Wording of Questions: 13 d. Ordering of Questions or Words: e. Type of Question: f. Data Entry Error: Example 1: A survey has bias. Determine the type of bias and suggest a remedy. 1. The village of Oak Lawn wishes to conduct a study regarding the income level of households within the village. The village manager selects 10 homes in the southwest corner of the village and sends an interviewer to the homes to determine household income. 2. Suppose you are conducting a survey regarding the sleeping habits of students. From a list of registered students, you obtain a simple random sample of 150 students. One survey questions is “How much sleep do you get?” 3. Cold Stone Creamery is considering opening a new store in O’Fallon. Before opening the store, the company would like to know the percentage of households in O’Fallon that regularly visit an ice cream shop. The market researcher obtains a list of households in O’Fallon and randomly selects 150 of them. He mails a questionnaire to the 150 households that asks about ice cream eating habits and flavor preferences. Of the 150 questions mailed, 4 are returned. 4. A health teacher wishes to do research on the weight of college students. She obtains the weights for all the students in her 9 am class by looking at their driver’s licenses or state ID’s. 5. A textbook publisher wants to determine what percentage of college professors either require or recommend that their students purchase textbook packages with supplemental materials, such as study guides, digital media, and online tools. The publisher sends out surveys by e-mail to a random sample of 320 faculty members who have registered with its Web site and have agreed to receive solicitations. The publisher reports that 80% of college professors require or recommend that 14 their students purchase some type of textbook package. 6. To determine the public’s opinion of the police department the police chief obtains a cluster sample of 15 census tracts within his jurisdiction and samples all households in the randomly selected tracts. Uniformed police officers go door to door to conduct the survey. Example 2: Read over the following scenarios and answer the question(s). 1. Order of Questions: Consider the following two questions: a. Do you believe that the government should or should not be allowed to prohibit individuals from expressing their religious beliefs at their place of employment? b. Do you believe that the government should or should not be allowed to prohibit teachers from expressing their religious beliefs in public school classrooms? Do you think the order in which the questions are asked will affect the survey results? If so, what can the pollster do to alleviate this response bias? Discuss the choice of the word prohibit in the survey questions. 2. Rotating Choices: Consider this question from a recent Gallup poll: Which of the following approaches to solving the nation’s energy problems do you think the U.S. should follow right now---(ROTATED: emphasize production of more oil, gas and coal supplies (or) emphasize more conservation by consumers of existing energy supplies)? Why is it important to rotate the two choices presented in the question? 3. Caller Id: How do you think caller ID has affected phone surveys? 4. Current Population Survey: In the federal government’s Current Population Survey, the response rate for 20- to 29-year olds is 85%; for individuals at least 70 years of age it is 99%. 15 Why do you think this is? Part 1: Getting the Information You Need Chapter 1: Data Collection Section 1.6: The Design of Experiments Learning Objective 1: Describe the characteristics of an experiment. Definition 16 An experiment is a controlled study conducted to determine the effect varying one or more explanatory variables or factors has on a response variable. Any combination of the values of the factors is called a treatment. In an experiment, the experimental unit is a person, object, or some other well- defined item upon which a treatment is applied. We often refer to the experimental unit as a subject when he or she is a person. A control group serves as a baseline treatment that can be used to compare to other treatments. A placebo is an innocuous medication, such as a sugar tablet, that looks, tastes, and smells like the experimental medication. Blinding refers to nondisclosure of the treatment an experimental unit is receiving. Definitions In single- blind experiments, the experimental unit (or subject) does not know which treatment he or she is receiving. In double-blind experiments, neither the experimental unit nor the researcher in contact with the experimental unit knows which treatment the experimental unit is receiving. Example 1: Caffeinated Sports Drinks Researchers conducted a double-blind, placebo-controlled, repeated-measures experiment to compare the effectiveness of a commercial caffeinated carbohydrate-electrolyte sports drink with a noncaffeinated carbohydrate- electrolyte sports drink and a flavored-water placebo. Sixteen highly trained cyclists each completed three trials of prolonged cycling in a warm environment: one while receiving the placebo, one while receiving the noncaffeinated sports drink, and one while receiving the caffeinated sports drink. For a given trial, one beverage treatment was administered throughout a 2-hour variable-intensity cycling bout followed by a 15-minute performance ride. Total work in kilogoules (kJ) performed during the final 15 minutes was used to measure performance. The beverage order for the individual subjects was randomly assigned. A period of at least 5 days separated the trials. All trials took place at approximately the same time of day in an environmental chamber at 28.5˚C and 60% relative humidity with fan airflow of approximately 2.5 meters per second (m/s). The researchers found that cycling performance, as assessed by the total work completed during the performance ride, was 23% greater for the caffeinated sports drink than for the placebo and 15% greater for the caffeinated sports drink than for the noncaffeinated sports drink. Cycling performances for the noncaffeinated sports drink and the placebo were nor significantly different. The researchers concluded that the caffeinated carbohydrate-electrolyte sports drink substantially enhanced physical performance during prolonged exercise compared with the noncaffeinated carbohydrate-electrolyte sports drink and the placebo. a. What does it mean for the experiment to be placebo- controlled? 17 b. What does it mean for the experiment to be double-blind? Why do you think it is necessary for the experiment to be double- blind? c. How is randomization used in this . experiment? d. What is the population for which this study applies? What is the sample? e. What are the treatments? f. What is the response variable? Learning Objective 2: Explain the steps in designing an experiment. Definition To design an experiment means to describe the overall plan in conducting the experiment. Conducting an experiment requires a series of steps. Step 1: Identify the Problem to Be Solved. The statement of the problem should be as explicit as possible and should provide the experimenter with direction. The statement must also identify the response variable and the population to be studied. Often, the statement is referred to as the claim. Step 2: Determine the Factors That Affect the Response Variable. The factors are usually identified by an expert in the field of study. In identifying the factors, ask, “What things affect the value of the response variable?” After the factors are identified, determine which factors to fix at some predetermined level, which to manipulate, and which to leave uncontrolled. Step 3: Determine the Number of Experimental Units. As a general rule, choose as many experimental units as time and money allow. Techniques exist for determining sample size, provided certain information is available. Step 4: Determine the Level of Each Factor. There are two ways to deal with the factors, control or randomize. 1. Control: There are two ways to control the factors. A. Set the level of a factor at one value throughout the experiment (if you are not interested in its effect on the response variable. B. Set the level of a factor at various levels (if you are interested in its effect 18 on the response variable). The combinations of the levels of all varied factors constitute the treatments in the experiment. 2. Randomize: Randomly assign the experimental units to various treatment groups so that the effect of factors whose levels cannot be controlled is minimized. The idea is that randomization averages out the effects of uncontrolled factors (explanatory variables). It is difficult, if not impossible, to identify all factors in an experiment. This is why randomization is so important. It mutes the effect of variation attributable to factors not controlled. Step 5: Conduct the Experiment. A. Randomly assign the experimental units to the treatments. Replication occurs, when each treatment is applied to more than one experimental unit. Using more than one experiment unit for each treatment ensures the effect of a treatment is not due to some characteristic of a single experimental unit. It is a good idea to assign an equal number of experimental units to each treatment. B. Collect and process the data. Measure the value of the response variable for each replication. Then organize the results. The idea is that the value of the response variable for each treatment group is the same before the experiment because of randomization. Then any difference in the value of the response variable among the different treatment groups is a result of differences in the level of the treatment. Step 6: Test the Claim. This is the subject of inferential statistics. Inferential statistics is a process in which generalizations about a population are made on the basis of results obtained from a sample. Provide a statement regarding the level of confidence in the generalization. Learning Objective 3: Explain the completely randomized design. Definition A completely randomized design is one in which each experimental unit is randomly assigned to a treatment. Example 2: Insomnia: Completely Randomized Design Researchers Jack D. Edinger and associates wanted to test the effectiveness of a new cognitive behavioral therapy (CBT) compared with both an older behavioral treatment and a placebo therapy for treating insomnia. They identified 75 adults with chronic insomnia. Patients were randomly assigned to one of three treatment groups. Twenty five patients were randomly assigned to receive CBT (sleep education, stimulus control, and time-in-bed restrictions), another 23 received muscle relaxation training (RT), and the final 25 received a placebo treatment. Treatment lasted 6 weeks, with follow-up conducted at 6 months. To measure the effectiveness of the treatment, researchers used wake time after sleep onset (WASO). Cognitive behavioral therapy produced larger improvements than did RT or placebo treatment. For example, the CBT-treated patients achieved an average 54% reduction in their WASO, whereas RT-treated and placebo-treated patients, respectively, achieved only 16% and 12% reductions in this measure. 19 Results suggest that CBT treatment leads to significant sleep improvements within 6 weeks, and these improvements appear to endure through 6 months of follow-up. a. What is the population being studied? b. What is the response variable in this study? c. What are the treatments? d. Identify the experimental units. e. Draw a diagram similar to Figure 7 or 8 to illustrate the design. Learning Objectiv Learning Objective 4: Explain the matched-pairs design. Definition A matched-pairs design is an experimental design in which the experimental units are paired up. The pairs are selected so that they are related in some way (that is, the same person before and after a treatment, twins, husband and wife, same geographical location, and so on).There are only two levels of treatment in a matched-pairs design. Example 3: Assessment To help assess student learning in her developmental math courses, a mathematics professor at a community college implemented pre- and posttests for her students. A knowledge-gained score was obtained by taking the difference of the two test scores. a. What is the response variable in this experiment? b. What is the treatment? 20 21

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