Statistics Concepts
The four main themes in a high school Advanced Placement (AP) Statistics course are exploring data, sampling, and experimentation, anticipating patterns, and statistical inference. Statistics is classified as a mathematics course and a common misunderstanding is that students spend most of their time computing data sets. However, most computations are done using graphing calculators or statistical software and only make up a small part of the curriculum. Many of the concepts and learning objectives in statistics require interpreting contexts and making data-based decisions regarding characteristics of populations. The following section provides an overview of key concepts addressed in the lesson activities.
Sampling
Samples from the same population can yield different results due to variability among the individuals of a population. Variability, the amount of variation or differences, of statistics is a central concept essential for understanding statistics. Controlling for variability when sampling a population is necessary for the sample data to provide statistically significant information that can be used to make inferences on the general population. Often it is not feasible or practical to take a census, collecting data from every individual in a population, sampling allows for insight regrading specific parameters, a characteristic of interest in a population. A simple random sample (SRS), a sample taken where every individual in a population has the same chance of being selected for the sample, is ideal as it also can help minimize bias.
Inference
Inference is when sample data are analyzed, and the results are used to make broader generalizations regarding a population. Practicing inference requires understanding the context of a problem, the parameters, and population of interest. Inference employs a variety of techniques including inference tests which utilize sample data to compute probabilities that are used to test assumptions. Sample data can also be used to construct confidence intervals, a range of expected amounts or observations of parameters within a population.
Hypotheses
Inference tests begin with a hypothesis, a claim that can be assessed. A null hypothesis, is the claim that there is no difference between specified populations. An alternative hypothesis, , the claim that there is a difference between specified populations. Inference tests have two possible outcomes. One outcome is that we reject the null hypothesis in support of the alternative hypothesis, meaning the sample data provided convincing statistical evidence that there is a difference between specified populations. The second outcome is that we fail to reject the null hypothesis, meaning the sample data does not provide convincing statistical evidence that there is a difference between specified populations. A challenge of statistics is that inference does not allow us to use samples of populations to prove there are no differences between populations.
Type I and Type II Errors
Samples provide insight regarding a specific parameter of a population. However, samples are not perfect data may be misleading. Errors do occur and have consequences. There are two types of errors that can occur when conducting statistical inference tests. Type I Error occurs if a statistical inference test fails to reject a null hypothesis when the null hypothesis is true. This means there is convincing evidence that the alternative hypothesis is true when it is in fact false. Type II Error is an error that occurs if a statistical inference test fails to reject a null hypothesis when the alternative is true. Here, the inference test concludes that there is not enough convincing evidence that the alternative hypothesis is true when it is in fact true.
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