Advanced High School Statistics: Third Edition

Each research question refers to a target population. In the first question, the target population is all swordfish in the Atlantic ocean, and each fish represents a case. Often times, it is too expensive to collect data for every case in a population. Instead, a sample is taken. A sample represents a subset of the cases and is often a small fraction of the population. For instance, 60 swordfish (or some other number) in the population might be selected, and this sample data may be used to provide an estimate of the population average and answer the research question.

We collect a sample of data to better understand the characteristics of a population. A variable is a characteristic we measure for each individual or case. The overall quantity of interest may be the mean, median, proportion, or some other summary of a population. These population values are called parameters. We estimate the value of a parameter by taking a sample and computing a numerical summary called a statistic based on that sample. Note that the two p’s (population, parameter) go together and the two s’s (sample, statistic) go together.

Two statistics we will study are the mean (also called the average) and proportion. When we are discussing a population, we label the mean as \(\mu\) (the Greek letter, mu), while we label the sample mean as \(\bar{x}\) (read as x-bar). When we are discussing a proportion in the context of a population, we use the label \(p\text{,}\) while the sample proportion has a label of \(\hat{p}\) (read as p-hat). Generally, we use \(\bar{x}\) to estimate the population mean, \(\mu\text{.}\) Likewise, we use the sample proportion \(\hat{p}\) to estimate the population proportion, \(p\text{.}\)

If these topics are still a bit unclear, don’t worry. We’ll cover them in greater detail in the next chapter.

Each conclusion is based on data. However, there are two problems. First, the data only represent one or two cases. Second, and more importantly, it is unclear whether these cases are actually representative of the population. Data collected in this haphazard fashion are called anecdotal evidence.

Anecdotal evidence typically is composed of unusual cases that we recall based on their striking characteristics. For instance, we may vividly remember the time when our friend bought a lottery ticket and won $250 but forget most the times she bought one and lost. Instead of focusing on the most unusual cases, we should examine a representative sample of many cases.

When we ask questions about the relationship between two variables, we sometimes also want to determine if the change in one variable causes a change in the other. Consider the following rephrasing of an earlier question about the county data set:

In this question, we are asking whether one variable affects another. If this is our underlying belief, then median household income is the explanatory variable and the population change is the response variable in the hypothesized relationship.

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Sometimes the explanatory variable is called the independent variable and the response variable is called the dependent variable. However, this becomes confusing since a pair of variables might be independent or dependent, so we avoid this language.

In many cases, the relationship is complex or unknown. It may be unclear whether variable \(A\) explains variable \(B\) or whether variable \(B\) explains variable \(A\text{.}\) For example, it is now known that a particular protein called REST is much depleted in people suffering from Alzheimer’s disease. While this raises hopes of a possible approach for treating Alzheimer’s, it is still unknown whether the lack of the protein causes brain deterioration, whether brain deterioration causes depletion in the REST protein, or whether some third variable causes both brain deterioration and REST depletion. That is, we do not know if the lack of the protein is an explanatory variable or a response variable. Perhaps it is both.

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https://www.nytimes.com/2014/03/20/health/fetal-gene-may-protect-brain-from-alzheimers-study-finds.html

There are two primary types of data collection: observational studies and experiments.

Researchers perform an observational study when they collect data without interfering with how the data arise. For instance, researchers may collect information via surveys, review medical or company records, or follow a cohort of many similar individuals to study why certain diseases might develop. In each of these situations, researchers merely observe or take measurements of things that arise naturally.

When researchers want to investigate the possibility of a causal connection, they conduct an experiment. For all experiments, the researchers must impose a treatment. For most studies there will be both an explanatory and a response variable. For instance, we may suspect administering a drug will reduce mortality in heart attack patients over the following year. To check if there really is a causal connection between the explanatory variable and the response, researchers will collect a sample of individuals and split them into groups. The individuals in each group are assigned a treatment. When individuals are randomly assigned to a group, the experiment is called a randomized experiment. For example, each heart attack patient in the drug trial could be randomly assigned into one of two groups: the first group receives a placebo (fake treatment) and the second group receives the drug. See the case study in Section 1.1 for another example of an experiment, though that study did not employ a placebo.