5A2 Fundamentals of Statistics [Filled) 8e

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Lansing Community College *

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119

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Mathematics

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Feb 20, 2024

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docx

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Math 119 5A Fundamentals of Statistics Student Learning Outcomes:  Select and use appropriate statistical methods to summarize and analyze data in contexts.  Develop and evaluate inferences and predictions that are based on data in context. Learning Objectives:  Define statistics, population, sample, population parameters, and sample statistics.  Describe 5 common sampling methods.  Define bias and explain how it can affect a statistical study.  Distinguish between observational studies, experiments, and retrospective studies.  For experiments, distinguish between a treatment group and a control group.  Understand the placebo effect and importance of anonymizing experiments.  Define and interpret a margin of error and confidence interval. Lead In question: In this section we will be discussing statistics and how we gather data. Humans have learned a lot about how to properly gather data in order to ensure our conclusions are valid. You do not want to gather data in such a way that might lead you to false results.  Statistics (singular) is the science of collecting, organizing, and interpreting data.  Statistics (plural) are the data that describe or summarize something. One of the things that we will discuss is the Placebo Effect. Check out the two videos below. BBC Documentary: https://www.youtube.com/watch?v=HqGSeFOUsLI Placebo Effect: https://www.youtube.com/watch?v=z03FQGlGgo0 Write out a sentence, explaining why someone who is collecting data or running an experiment, needs to be aware of the placebo effect. The placebo effect can lead us to incorrect conclusions if we are not aware of it. People perceive positive effects of treatments even when there might not be any, so we have to control for this when collecting data or running an experiment. I. How Statistics Works  The population in a statistical study is the complete set of people or things being studied (the what/who we are trying to learn about, it will often contain the word “all” when you describe it.)  The sample is the subset of the population from which the raw data are actually obtained (the what/who we have data on, it will often give the exact number.)

 Parameters are specific characteristics of the population that a statistical study is designed to estimate.  Statistics are numbers or observations that summarize the raw data; since the sample is what we have in our hands to work with, we can compute or crunch these. In general, we compute statistics from the sample in order to estimate the corresponding parameters from the population. To help with remembering this terminology, the “s’s for s ample s tatistic” go together and the “p’s for p opulation p arameter” go together. Example: Let’s say we wanted to study the LCC student body. a. This is our population . b. Are the students in our Math 119 class a sample? If so, are they a good sample? Yes, our Math 119 class is a subset (small collection) of the students who attend LCC. One section of Math 119, however, is probably not representative of the LCC student body as a whole since all of the people in our section may have certain traits in common that differ from the general population. (do we prefer online or face-to-face classes? Do we prefer to meet at certain times during the day? Which days do we prefer to meet? Etc.) In order to get a sample that accurately represents the entire student body, we would have to be strategic about how we selected the students in our sample. c. Will we be able to extend what we learn from the sample to the population in this case? We cannot generalize our results to the entire LCC student body since our sample does not represent the student body as a whole. However, our sample might be useful to say something about Math 119 students who prefer online or face-to-face classes. Example: In order to gauge public opinion on the Presidents’ plan to contain Iran’s nuclear program, the Pew Research Center surveyed 1001 Americans by telephone to find the percentage that favor the plan. a. Identify the population and sample. Since it is the sample that is typically described more in the words of every problem, I would suggest you start with that…

Sample: the 1001 Americans surveyed by telephone (that’s what/who we have data on and thus in our hands so to say that we can work with) Population: the set of all Americans (the what/who we are trying learn about) b. Describe in words the population parameters, and sample statistics. The sample statistic is the percentage of the 1001 American’s surveyed (our sample) who favor the Presidents’ plan. We have the data and we would be able to compute this percentage. The parameter would be something like the percentage of all Americans who favor the Presidents’ plan. We would take the sample statistic and use it as an estimate for the whole population. Note that the parameter and statistics are always the same type of number (percentage, average, etc.). II. Choosing a Sample If the sample fairly represents the population as a whole, then it’s reasonable to make inferences from the study. If the sample is NOT representative, then there’s little hope of drawing accurate conclusions about the population.  A representative sample is a sample in which the relevant characteristics of the sample members match those of the population. Common Sampling Methods  Simple random sampling: We choose a sample of items in such a way that every sample of the same size has an equal chance of being selected.  Systematic sampling: We use a simple system to choose the sample, such as selecting every 10th or every 50th member of the population.  Convenience sampling: We choose a sample that is convenient to select, such as people who happen to be in the same classroom.  Cluster Sampling: We first divide the population into groups, or clusters , and select some of these clusters at random. We then obtain the sample by choosing all the members within each of the selected clusters.  Stratified sampling: We use this method when we are concerned about differences among subgroups, or strata, within a population. 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.

Usually, it is “cluster” and “stratified” that are the most difficult for students to distinguish. In both of them, we group the population. But in cluster, we take some of the groups. In stratified, we take some members of every group. The study can be successful only if the sample is representative of the population. Sample size is important, because a large well-chosen sample has a better chance of being representative than a small one. However, the selection process is even more important: A small well-chosen sample is likely to give better results than a large poorly chosen sample. Example: Identify the type of sampling in each of the following cases. a. You are conducting a survey of students in a dormitory. You choose your sample by knocking on the door of every fifth room. Systematic b. To survey opinions on a proposed new water line, a research firm randomly draws the addresses of 200 homeowners from a public list of all homeowners. Simple Random

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