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I have this homework in MA-132 course I have the questions and the answers are in the slides that are attached you only need to put the answers. chapter_1_definitions.pdf ma_132_section_1.1_slides.pdf ma_132_section_1.2_slides.pdf ma_132_section_1.3_and_1.4_slides.pdf ma_132_section_1.5_slides.pdf Unformatted Attachment Preview MA 132: Chapter 1 Definitions Section 1.1: Introduction to the Practice of Statistics • Population: The complete set of people or things being studied • Sample: The subset of the population from which information is actually obtained • Parameter: A numerical characteristic of a population. (Usually not measured and/or not measurable) • Statistic: A numerical summary of a sample or (a number that summarizes the raw data) • Descriptive Statistics: Organizing and summarizing data • Inferential Statistics: Try to infer reasonable values for population parameters (not measured) from the sample statistics (measured) • Quantitative: Numerical measures of an individual on which meaningful arithmetic operations can be performed on the values of the variable. • Qualitative: Non-numerically measurable attributes or characteristics • Response Variable: The focus of a question in a study or experiment • Explanatory Variable: A variable that explains changes in the response variable Section 1.2: Observational Studies versus Designed Experiments • Observational Studies: The researcher observes the behavior of the individuals without attempting to influence the outcome of the study • Experiments: The 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 • Confounding: Occurs when the effects of two or more explanatory variables are not separated • Lurking Variable: An explanatory variable that was not considered in a study, but that affects the value of the response variable in a study Section 1.3: Simple Random Sampling • Simple Random Sample (SRS): A random sample where every sample of size n has the same chance of being selected from the population of size N . There can be no subgrouping of the population. Section 1.4: Other Effective Sampling Methods • Stratified: Population is divided into two or more non-overlapping groups classed strata and then a simple random random sample is drawn from each strata. • Cluster: Population is divided into clusters and then entire clusters are randomly selected. • Convenience: A sample in which the individuals are easily obtained and not based on randomness. • Systematic: Every nth member of the population is selected. 2 Section 1.5: Bias in Sampling • If the results of the sample are not representative of the population, then the sample has bias. • Sampling bias: means that the technique used to obtain the sample favors one part of the population over another. – e.g., undercoverage occurs when the proportion of one segment of the population is lower in the sample than it is in the population. • Nonresponse bias: occurs when individuals in sample who do not respond to the survey have different opinions from those who do. • Response bias: occurs when answers on a survey do not represent the true feelings of the respondent. – Interviewer error – Misrepresented answers (e.g., people may overestimate their abilities) – Question wording (deliberate, unintentional, or unnecessary complexity) – Question ordering • Types of Errors – Nonsampling errors: result from sampling bias, nonresponse bias, response bias, or data-entry error. Such errors could also be present in a complete census of the population. – Sampling errors: result from using a sample to estimate information about a population. This type of error occurs because a sample gives incomplete information about a population. 3 Section 1.6: The Design of Experiments • Experiments: Vocabulary – Subjects (or Experimental units): individuals studied in an experiment. – Factors: the explanatory variables in an experiment. – Treatment: any specific experimental condition applied to the subjects; if there are several factor, a treatment is a combination of specific values of each factor. • Experiments: Basic Principles – Comparative: compare more than one treatment at a time–to ensure realism. – Randomization: to balance out lurking variables across treatments. – Control group: serves as a baseline treatment that can be used to compare to other treatments–to account for changes not related to the treatment of interest. – Placebo: an innocuous medication, such as a sugar tablet, that looks, tastes, and smells that the experiment medication–to control for the power of suggestion. – Blinding: nondisclosure of the treatment an experimental unit is receiving. – Single-blind: the subject does not know which treatment they are receiving. – Double-blind: neither the researcher nor the subject know which treatment they are receiving. • Experimental Designs – Completely randomized design (CRD): each experimental unit is randomly assigned to a treatment. – Matched-pairs design: choose pairs of subjects that are as closely matched as possible and then randomly assign one treatment to one subject and the second treatment to the other subject. Sometimes a “pair” could be a single subject receiving both treatments and the order of treatments would be randomized for each subject. 4 MA 132 Section 1.1 Introduction to Statistics Nielsen Ratings n n n For the week of April 27, 2020, Young Sheldon was the top-rated show on broadcast TV in the U.S. Who counted? Did they ask every person in the U.S.? Terminology n n n n n Population: for this study of TV viewing in the U.S., the population is all Americans. Generally, the population is the complete set of people or things being studied Sample: the subset of the population from which information is actually obtained Raw data: the information that is collected from the sample Statistic: a numerical summary of a sample or (a number that summarizes the raw data) Parameter: a numerical characteristic of a population. (Usually not measured and/or not measureable) Statistics n n n Descriptive Statistics: organizing and summarizing data (6.2\% of the viewers in the Nielsen sample watched Young Sheldon) Inferential Statistics: Try to infer reasonable values for population parameters (not measured) from the sample statistics (measured). Inferential Statistics usually includes a level of confidence. (We are 95\% confident that between 4.7\% and 7.7\% of viewers watched Young Sheldon) Parameter vs. Sample n n n n Parameter: 48.2\% of all students on Marymount’s campus own a car. This is a numerical summary of a POPULATION Statistic: A sample of 100 students is obtained, and from the sample, 46\% own a car. This is a STATISTIC because it is a numerical summary of a sample. Examples n Only 12 men have walked on the moon. The average age of these men at the time of their moonwalks was men that have population all walked 39.96 years. -_ q n Parameter the moon In a national survey about driving habits, 79\% of respondents admitted to frequently driving over the posted speed limit. - q n on statistic The average score on the final for a class of 17 statistics population class of students was 73\%. students 17 = q Parameter Variables n n n n n n Variables are the characteristics of the individuals within the population. Variables can be classified into two groups. Quantitative Variables – numerical measures of an individual on which meaningful arithmetic operations can be performed on the values of the variable. Examples: height, temperature, number of days studied last week Qualitative Variables – non-numerically measurable attributes or characteristics ( categories) Examples: hair color, gender, zip code, the number on a football player’s jersey student ID numbers , Handout n Answer the Section 1.1 Questions MA 132 Section 1.2 Observational Studies versus Designed Experiments The Process of Statistics n n n n (1) Identify the research objective (2) Collect the data needed to answer the question(s) posed in (1) (Chapter 1) (3) Describe the data (Descriptive statistics allow for an overview of the data and help determine the type of statistical methods that should be used – Chapters 2 – 4) (4) Perform inference (Apply appropriate techniques to extend the results obtained from the sample to the general population and report a level of reliability of the results – Chapters 9 – 12) Example n CBS News and the NYT conducted a poll in Sept. 2014 and asked “Do you favor or oppose raising the minimum wage from $7.25 to $10.10 per hour?” Step 1 n n n Identify Research Objective: The researchers wanted to determine the percentage of adult Americans who favor raising the minimum wage. The POPULATION being studied was adult Americans Step 2 n n n Collect the data needed to answer the question in Step 1 It is unreasonable to expect to survey all 200 million adult Americans to determine how they feel about the minimum wage. Researchers surveyed a sample of 1009 adult Americans. Of those surveyed, 706 stated they favor an increase in the minimum wage to $10.10 per hour Step 3 n n n Describe the data Of the 1009 individuals in the study, 70\% (706/1009) believe the minimum wage should be raised to $10.10 per hour. This is a DESCRIPTIVE STATISTIC because it is a numerical summary of the data. Step 4 n n n n n Perform Inference CBS News and the NYT wanted to extend the results of the survey to all adult Americans. When generalizing a result from a sample to the general population, results are uncertain. Researchers reported a 3\% margin or error. This means CBS News and the NYT were 95\% certain that the percentage of ALL americans who favor raising the minimum wage to $10.10 is between 67\% (70 – 3) and 73\% (70 + 3). Observational Studies n n n The researcher observes the behavior of the individuals without attempting to influence the outcome of the study. Example: 791,710 middle-aged women in the UK were followed over a period of 7 years. During this time, 1261 incidences of brain tumors in the group were recorded. Researchers compared women who had never used mobile phones with ones who had, and found no significant difference in the incidence of brain tumors between the two groups. Designed Experiments n n n n n n n The 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. Example: Researchers randomly divided 480 rats into three groups. Rats in group 1 were exposed to the amount of RF radiation emitted by analogue cell phones Rats in group 2 were exposed to the amount of RF radiation emitted by digital cell phones Rats in group 3 received no radiation Exposure was done 4 hours a day, 5 days a week for 2 years. After 505 days of exposure, the researchers found no difference in rates of any tumor type. In both studies… n n The explanatory variable is the level of cell phone usage. The response variable is the incidence of brain tumors. Observational Study Example n n n n n n Researchers wanted to determine the long-term benefits of the flu shot on seniors aged 65 years and older by looking at records of over 36,000 seniors for 10 years. The seniors were divided into 2 groups, those who chose to get flu shots and those who did not. After looking at the records, it was determined that seniors who get flu shots are 27\% less likely to be hospitalized for pneumonia or flu, and 48\% less likely to die from pneumonia or flu. Are there any potential confounding variables? Could it be that seniors who choose to get a flu shot are more health conscious in general, or are able to get to the clinic more easily (and are therefore already healthier or better off financially?) Does race, income, or gender play a role in whether one might contract (and possibly die from) the flu? Confounding n n n n n 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. Often, the cause of confounding is a lurking variable. A lurking variable is an explanatory variable that was not considered in a study, but that affects the value of the response variable in a study. A confounding variable is an explanatory variable that was considered in a study, whose effect cannot be distinguished from a second explanatory variable. Are experiments always better? n n n n n Designed experiments may allow researchers to identify cause and effect relationships. But, it is not always possible to use a designed experiment Sometimes observational studies are the only ethical and/or practical way to get the data They have the advantage of costing less, being quicker to run, and can be utilized with a greater range of individuals. However, observational studies do not allow a researcher to claim causation, only association Example n Incoming freshman are randomly divided among 10 sections of EN 101. Five of the sections are taught using traditional techniques, and the other five are taught using a reform method. After 1 year, each group is given an achievement test to compare proficiency. q Is this a designed experiment or an observational study? • q q q What is the explanatory variable? teaching Is the explanatory variable• qualitative or quantitative? What is the response variable? score q q technique on achievement test Is the response variable qualitative or quantitative? • List any lurking variables that may have affected the results of this study. Example n Researchers asked 93,676 women to disclose their coffee drinking habits and also determined which of the women had non-melanoma skin cancer. 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. q Is this a designed experiment or an observational study? - q q q What is the explanatory variable? how much coffee they drink day Is the explanatory variable qualitative or0 quantitative? What is the response variable? whether or not they q q had per skin cancer Is the response variable o qualitative or quantitative? List any lurking variables that may have affected the results of this study. Handout n Answer the Section 1.2 Questions MA 132 Sections 1.3 & 1.4 Sampling Techniques Nielsen Ratings n n n For the week of April 27, 2020, Young Sheldon was the top-rated show on broadcast TV in the U.S. Who counted? Did they ask every person in the U.S.? Choosing your sample n n n If 6.2\% of the viewers in the Nielsen sample watched Young Sheldon, is it reasonable to assume that 6.2\% of all Americans watched the show? Depends on how accurately the sample represents the entire population. For the results of a survey to be reliable, the characteristics of the individuals in the sample must be representative of the characteristics of the individuals in the population. Bad Samples n n Suppose you want to know the proportion of adult Americans who consider themselves to be baseball fans. If you conducted a survey outside of Nationals Park (home of the Washington Nationals baseball team), are your results likely to be reliable? Bad Samples Continued n n n n Suppose you want to know the proportion of MU students who work. Could we get reliable results by surveying our statistics class? Do the students in this class mirror the proportion of freshmen, sophomores, juniors, and seniors of the general student body? Does the proportion of men and women in the class mirror the proportion of men and women in the general population? Simple Random Sample n n 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 How to choose a SRS? n n n n Put the names of the N individuals into a hat and choose n of them (this is not always practical) Use Technology (calculator or computer) Before obtaining a SRS, we need a FRAME. A frame is a list of all the individuals in a population. Other Effective Sampling Techniques n n n Stratified Systematic Cluster Common Sampling Methods n Stratified Sampling: We first identify the subgroups and then draw a simple random sample within each subgroup. The total sample consists of all the samples from the individual subgroups. Obtaining a Stratified Sample n n n The president of a university wants to conduct a survey to determine the community’s opinion regarding campus safety. The campus community is divided into 3 groups: resident students, non-resident students, and staff. A simple random sample is obtained from each group. Common Sampling Methods n Cluster Sampling – divide an area into sections, randomly select some sections and choose every member in the selected sections Obtaining a Cluster Sample n n A sociologist wants to gather data regarding household income within the city of Boston. The city of Boston can be set up so that each city block is a cluster. Once the city blocks have been identified, obtain a SRS of the city blocks and survey all households on the selected blocks. Common Sampling Methods n Systematic Sampling: We use a system to choose the sample such as selecting every 5th or every 50th member of the population Common Sampling Methods n Convenience Sampling: We use a sample that is convenient to select, such as people who happen to be in the same classroom. Do you support the death penalty? Examples: Identify the sampling technique used. n A $50 gift certificate is given away at the Annual Banker’s Convention. Tickets are placed in a bin, and the tickets are mixed up. Then the winning ticket is selected by a blindfolded person. q n random Children in a large city are classified based on the neighborhood school they attend. A random sample of five schools is selected. All the children from each selected school are included in the sample. q n simple cluster A radio station asks its listeners to call in their opinion regarding the use of U.S. forces in peacekeeping missions. q convenience Examples: Identify the sampling technique used. n The first 50 people entering a zoo are asked if they support an increase in taxes to support zoo expansion. q n n convenience An owner of a grocery store decides to conduct a survey of her employees regarding worker morale. She randomly chooses 5 managers and 20 hourly employees strata → managers to survey. → stratified q hourly Students a Portland State University are classified according to their major, then a random sample of 15 students from each major is selected. q stratified strata - majors Handout n Answer the Section 1.3 and 1.4 Questions MA 132 Section 1.5 Bias in Sampling Bias n n n If the results of the sample are not representative of the population, then the sample has BIAS There are 3 sources of bias in sampling Sampling, Nonresponse, Response Sampling Bias n n The technique used to obtain the individuals in the sample tends to favor one part of the population over another. Any convenience sample has sampling bias because the individuals are not chosen through a random sample. Sampling Bias Example n n n n n Have you ever heard of Alf Landon? How about Franklin Roosevelt? Literary Digest predicted in 1936, based on a poll, that Alf Landon would win the presidency in a landslide over Franklin Roosevelt What was their sampling method? Was the sample representative? What was the frame? n n n n Literary Digest used a list of their subscribers, telephone directories, and automobile owners as their frame This election took place during the height of the Great Depression In 1936, most subscribers to magazines, owners of telephones and cars were Republicans The sample had an undercover ... Purchase answer to see full attachment