AP Statistics College Board

This subject is broken down into 79 topics in 9 modules:

  1. Collecting Data 7 topics
  2. Exploring One-Variable Data 10 topics
  3. Exploring Two-Variable Data 9 topics
  4. Inference for Categorical Data: Chi Square 7 topics
  5. Inference for Categorical Data: Proportions 10 topics
  6. Inference for Quantitative Data: Means 10 topics
  7. Inference for Quantitative Data: Slopes 6 topics
  8. Probability, Random Variables, and Probability Distributions 12 topics
  9. Sampling Distributions 8 topics
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This page was last modified on 28 September 2024.

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Statistics

Collecting Data

Inference and Experiments

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Inference and Experiments

Inference and Experiments

In understanding "Inference and Experiments" as part of the data collection process, the key areas to focus on are: randomness, sampling, experimentation, and observational studies.

Randomness

  • Randomness is the foundation of statistical inference and is vital to ensure that data collected is representative of the entire population.
  • The process of engaging random selection or random assignment helps to eliminate bias, ensuring the sample represents the entire population.
  • Random selection involves selecting individuals to participate in a survey or study at random.
  • Random assignment is employed when each individual has an equal chance of being allocated to any group in an experiment.

Sampling

  • Sampling is the process of selecting a subset of individuals from the population, allowing for the collection of data.
  • Simple random sampling is when each individual has an equal probability of being chosen.
  • When groups, or clusters, of individuals are selected, this is known as cluster sampling.
  • Stratified sampling splits the population into groups (or strata) and then selects random samples from each strata.
  • Systematic sampling involves selecting every nth |individual from the population.

Experimentation

  • In experimentation, researchers actively control and manipulate the variables to determine their effects.
  • An experiment must include at least two conditions: the treatment condition and the control condition.
  • Control group usually receives no treatment or a placebo treatment, while the experimental group receives the treatment being tested.
  • In a randomised controlled trial, participants are randomly assigned to control or treatment groups to reduce bias.
  • Blinding is a process in which subjects do not know whether they are in the control or treatment group. Double-blinding extends this process to researchers.

Observational Studies

  • In an observational study, researchers observe and measure variables of interest without actively controlling or manipulating them.
  • These studies can be retrospective (looking at historical data) or prospective (collecting new data going forward).
  • In a case-control study, two existing groups differing in outcome are identified and compared on the basis of some supposed causal attribute.
  • Cohort studies identify a group of individuals to study over time.

It is crucial to remember that both randomness and bias are key components to consider while collecting data for statistical analysis. Understanding the differences between sampling techniques, and the specifics of experimentation and observational studies will give the best chance to pull accurate inferences from the data.

Course material for Statistics, module Collecting Data, topic Inference and Experiments

Statistics

Inference for Categorical Data: Proportions

Potential errors when performing tests

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Potential errors when performing tests

Types of Errors in Hypothesis Testing

  • Two main types of errors can occur in hypothesis testing: Type I errors and Type II errors.

Type I Error

  • A Type I error occurs when the null hypothesis is true, but we mistakenly reject it.
  • In other words, we think we found a significant effect, but it was actually due to random chance.
  • The probability of making a Type I error is denoted by α, also known as the significance level.
  • Reducing α reduces the chances of making a Type I error, but it also makes it harder to detect a true effect when one exists.

Type II Error

  • A Type II error occurs when the null hypothesis is false, but we fail to reject it.
  • In other words, a significant effect exists, but we missed finding it.
  • The probability of making a Type II error is denoted by β.
  • Improving the study design, such as increasing the sample size or using more reliable measurement tools, can reduce β and increase the statistical power, which is the probability of correctly rejecting a false null hypothesis.

Errors in Proportions

Sampling Error

  • A sampling error occurs when a sample proportion does not accurately represent the population proportion.
  • It is due to the inherent randomness in sampling. Each sample drawn from the same population can produce slightly different results.
  • A larger sample size generally reduces the sampling error and provides a more accurate estimate of the population proportion.

Non-Sampling Errors

  • Non-sampling errors arise from factors other than the sampling process. They can be due to errors in data collection, response bias, or measurement errors.
  • For example, a poorly designed survey question might lead to response bias, or a malfunctioning measuring device might produce inaccurate readings, leading to measurement errors.

Understanding Data Assumptions

  • Inferential statistics, including tests of proportions, often rely on certain assumptions about the data.
  • Violating these assumptions can lead to inaccurate or misleading results.
  • For instance, the test for comparing two proportions assumes that the samples are independent of each other. If this assumption is violated, the test might produce invalid results.
  • Always check the assumptions of the statistical methods you are using and ensure your data meets these assumptions before proceeding with hypothesis testing. Violations of assumptions may necessitate the use of a different statistical method.

Course material for Statistics, module Inference for Categorical Data: Proportions, topic Potential errors when performing tests

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