Chapter 5: Statistical Reasoning Section 5A Fundamentals of Statistics Statistics is the science of collecting, organizing and interpreting data Statistics is the data that describe or summarize something Population: Sample: Population parameters: Sample statistics: Example 1: Population and Sample For each of the following cases, describe the population, sample, population parameters, and sample statistics. a) Agricultural inspectors for Jefferson County measure the levels of residue from three common pesticides on 25 ears if corn from each of 104 cornproducing farms in the county. b) Anthropologists determine the average brain size of early Neandertals in Europe by studying skulls found at three sites in southern Europe
Basic steps in a statistical study: 1) Identify goals (determine the population and what you d like to learn) 2) Draw from population (choose a sample) 3) Collect raw data and summarize (find sample statistics) 4) Make inferences about population (using the sample statistics) 5) Draw conclusions (what did you learn and did you achieve your goal) Example 2: Unemployment Survey Each month, the U.S. Labor Department surveys 60,000 households to determine characteristics of the U.S. work force. One popular parameter of interest is the U.S. unemployment rate, defined as the percentage of people who are unemployed among all those who are either employed or actively seeking employment. Describe how the five basic steps of a statistical study apply to this research. Choosing a sample: This may be the most important step in any statistical study. If the sample fairly represents the population as a whole, then it s reasonable to make inferences from the sample to the population. Representative sample: Common sampling methods Simple random sampling: every sample of a given size has an equal chance of being selected Systematic sampling: simple system such as selecting every 10 th person or every 50 th member of the population Convenience sampling: convenient to select such as people who happen to be in the same classroom (use the results that are readily available) Stratified sampling: use this when you are concerned about differences among subgroups (strata) within a population. Identify subgroups, then draw a simple random sample within each subgroup. The total sample consists of all the samples from the individual subgroups.
Example 3: Sampling Methods Identify the type of sampling used in each of the following cases, and comment on whether the sample is likely to be representative of the population. a) You are conducting a survey of students in a dormitory. You choose your sample by knocking on the door of every 10 th room. b) To survey opinions on a possible property tax increase, a research firm randomly draws the address of 150 homeowners from a public list of all homeowners. c) Agricultural inspectors for Jefferson County check the levels of residue from three pesticides on 25 ears of corn from each of the 104 cornproducing farms in the county. d) Anthropologists determine the average brain size of early Neandertals in Europe by studying skulls found at three sites in southern Europe. Watch out for Bias Bias: Two Basic Types of Statistical Study 1) In an observational study, researchers observe or measure characteristics of the sample members but do not attempt to influence or modify these characteristics. 2) In an experiment, researchers apply a treatment to some or all of the sample members and then look to see whether the treatment has any effects. In experimental treatments, create two groups of people: a treatment group and a control group. Treatment group: Control group:
The Placebo Effect and Blinding For experiments involving people, using a treatment and control group may not be enough to get reliable results. People are affected by beliefs and real treatments. A placebo lacks the active ingredients of a treatment being tested in a study, but it is identical in appearance to the treatment. Participants cannot distinguish between the placebo and the real treatment. The placebo effect refers to the situation in which patients improve simply because they believe they are receiving a useful treatment. Blinding: Single-blind: Double-blind: Example 4: What s wrong with this experiment? For each of the experiments described below, identify any problems and explain how the problems could have been avoided. a) A chiropractor wants to know if his adjustments relieve back pain. He performs adjustments on 25 patients with back pain. Afterward, 18 of the patients say they feel better. He concludes that the adjustments are an effective treatment. b) A new drug for attention deficit disorder (ADD) is supposed to make the affected children more polite. Randomly selected children suffering from ADD are divided into treatment and control groups. Those in the control group receive a placebo that looks just like the real drug. The experiment is single-blinded. Experimenters interview the children one-on-one to decide whether they became more polite.
A case-control study (retrospective study) is an observational study that resembles an experiment because the sample naturally divides into two or more groups. The participants who engage in the behavior under study form the cases, which makes them like a treatment group in an experiment. The participants who do not engage in the behavior are the controls, making them like a control group in an experiment. These studies are used when it is impractical or unethical to conduct an experiment. Example: A study about how alcohol consumed during pregnancy affects newborn babies. It is unethical to force pregnant mothers to consume alcohol. Case-control- participants naturally form groups by choice Cases consist of mothers who consume alcohol during pregnancy by choice Controls consist of mothers who choose not to consume alcohol during pregnancy Example 5: Which Type of Study? For each of the following questions, what type of statistical study (observational, experiment, case-control) is most likely to lead to an answer? Why? If experiment or case-control, identify the groups within the study. a) What is the average income of stockbrokers? b) Do seat belts save lives? c) Can lifting weights improve runners times in a 10-km race? d) Can a new herbal remedy reduce the severity of colds?
Surveys and Opinion Polls These may be the most common types of statistical study. Results include a margin of error. Sample statistic: Confidence Interval: Margin of error: How confident can we be in a poll result? Unless we are told otherwise we assume that the margin of error is defined to give us 95% confidence that the confidence interval contains the population parameter. This idea will be discussed in greater detail in Chapter 6. Example 6: Close Election An election eve poll finds that 52% of surveyed voters plan to vote for Smith, and she needs a majority (more than 50%) to win without a runoff. The margin of error in the poll is 3 percentage points. Will she win?