Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11STATS 250 NOTES* (condensed) Exam 1Chapter 1**Report decimal places 3-4 places after the decimal point.**Stats investigation process --- ADEUFR:1. Ask a Research Question2. Design a study and collect data3. Explore the data, provide graphical displays, and numerical summaries4. Use statistical analysis methods to draw inferences from the data5. Formulate conclusions, communicate findings and answer research questions6. Reflect and look forward(point out limitations and suggest further studies)When data is not collected for question --- ITEUFR:1. Import the data2. Tidy the data3. Explore the data, provide graphical displays, and numerical summaries4. Use statistical analysis methods to draw inferences from the data5. Formulate conclusions, communicate findings and answer research questions6. Reflect and look forward(point out limitations and suggest further studies)Categorical Variables --- or qualitative variables●place an individual or item into one of the several groups or categories, which are calledlevels.● Nominal variables vs. Ordinal variables, ordinal follows specific order(small, medium,large)● They report frequency counts●Relative frequency - Decimals or Percent●Graphs used to display -- bar charts, pie chart● Example: Ice cream flavors are a categorical variable, and how many flavors are thelevels, and how many people like which flavor would be frequency counts, and when youreport the counts in decimals or percentages its considered relative frequencyNumerical Variables --- or quantitative or measurement●variables that take on a wide range of numerical values, and it is sensible to do math withnumerical variables.●Discrete: numerical variables with jumps. Ex: 1, 2, 3, 4, 5● Continuous variables: Can take any value in an interval or collection of intervals. Ex: 2.3,6.7. 7.808, 8.9999● Numerical variables can have subtypes. Ex: ages 18-25, 25-34, 35-44…●It’s possible to turn numerical values to categorical by grouping them, but not categoricalvalues to numerical●Graphs: Histograms, boxplots, scatterplotsContingency tables: one-way data is summarized with tablesData Matrix: Common way to organize raw, unprocessed data. They have columns and rows.Population: The entire group we are interested in learning about. All undergraduate students inthe US.● We dont observe every case in a study. Would be time-consuming and costs too much,and sometimes can destroy the item in the process of measurementSample: A subset of the cases that is often a small fraction of the overall population.Undergraduate students un UMICH●It provides an estimate for the overall population, less time-consuming, and less costly.●There could be biases in a sample so the way we sample is important● Biases: Convenience sampling, Response bias, Non-response biasAnecdotal Evidence: Typically composed of unusual cases that are recalled based on theirstriking characteristics.Sampling from a Population: To draw inferences about a population1. The sample must be representative of the entire population2. Use random sampling: subjects of study/experiment should be selected randomly toensure the sample is representativeExplanatory and Response variables●Explanatory: a variable that predicts the outcome● Response: a variable that is the outcome --- responds to the explanatory variableTwo types of data collection: Observational and Experimental●Observational Studies: refer to instances where researchers collect data in a way that doesnot directly interfere with how the data arise. They simply observe.○ Interested in looking at the relationship between two or more variables○ Data is usually collected only by monitoring what occurs○Making causal inferences based on observational studies is difficult but notimpossible●Experiments: researcher directly influences the process by which data arise.○ Subjects are usually assigned to one or more treatments RANDOMLY○ There is usually a control variable or a placebo effect○Require the primary explanatory variable in a study be assigned to each subject byresearchers○Making causal conclusions is reasonable, depending on the way the explanatoryvariable is assigned.Observational StudiesSimple Random Sample (SRS)●Of n observations from a population is one in which each possible sample of that size hasthe same chance of being in the sample that is selected●So, every member/case in a population has an equal chance of being included and there isno implied connection between members/cases in the sample.●The best way of ensuring that the sample is representative of the population it is chosenfrom.● For experiments: Makes groups similar as possible(dispersing confounding factors evenlybetween groups) with only difference being due to the treatment.Stratified Sampling●Strata: Group of individuals or cases in a population who share characteristics thought tobe associated with the variable we want to measure●A “divide and conquer” sampling strategy● The population is divided into non-overlapping strata● Each SRS is taken from each stratum●Works when there is variability between each stratum but not much variability withineach stratumConvenience Sampling●Refers to samples that are obtained by measuring whatever or whoever is available to bemeasured. So nearest cases and subjects --- convenient●Rarely representative of a larger population● The sampling method is often biasedBIAS in sampling:●Results obtained based on a survey are biased if the method used to obtain those resultswould consistently produce values that are either too high or too low.● Selection Bias: occurs when the method for selecting participants produces a sample thatdoes not represent the population of interest. Ex: convenience sampling● Nonresponse Bias: occurs when a representative sample is chosen for a survey, but asubset cannot be contacted or does not respond. Ex: Voluntary response sample, orsurveying only a specific people and not the entire population.● Response Bias: Occurs when participants respond differently from how they truly feel.The way questions are worded, the way the interviewer behaves, as well as many otherfactors, might lead an individual to provide false information●Sampling Bias: A type of bias that occurs when the method for selecting participantscauses some individuals in the population to more or less likely to be included in thesample