Inference and Experiments - AP Statistics

1. What is the effect of increasing sample size on margin of error?

   Decreases the margin of error. Larger samples provide more precise estimates.

2. Define 'standard deviation'.

   Measure of data dispersion around the mean. Quantifies how spread out data points are.

3. What is a 'confidence interval'?

   Range of values likely to contain a population parameter. Provides an estimate with associated uncertainty.

4. What is a Type I error?

   Rejecting a true null hypothesis. A false positive error with probability $\alpha$.

5. What is the 'central limit theorem'?

   Distribution of sample means approximates normality as sample size increases. Foundation for statistical inference with large samples.

6. What is 'stratified sampling'?

   Dividing population into subgroups and sampling each proportionally. Ensures all subgroups are adequately represented.

7. What does 'random sampling' ensure?

   Each member of the population has an equal chance of selection. Eliminates selection bias and enables valid inference.

8. What is the 'law of large numbers'?

   As sample size increases, sample mean approaches population mean. Foundation for probability and statistical inference.

9. What is a Type II error?

   Failing to reject a false null hypothesis. A false negative error with probability $\beta$.

10. State the significance level symbol.

    $\text{alpha}$ or $\text{α}$. Greek letter alpha represents the significance threshold.

11. Define 'observational study'.

    Study observing subjects without manipulation or intervention. Cannot establish causation, only association.

12. Define 'power of a test'.

    Probability of correctly rejecting a false null hypothesis. Power equals $1 - \beta$ where $\beta$ is Type II error rate.

13. State one advantage of stratified sampling.

    Ensures representation of all subgroups in the sample. Particularly useful when subgroups differ significantly.

14. Which measure indicates the center of a data set?

    Mean or median. Describes the central tendency of the distribution.

15. Define a 'control group' in an experiment.

    A group that does not receive the treatment, used for comparison. Provides a baseline to measure treatment effects against.

16. What does 'external validity' refer to?

    Extent to which results generalize to other contexts. Relates to the generalizability of study findings.

17. Which test is used for comparing means of two groups?

    $t$-test. Uses the $t$-distribution when population standard deviation is unknown.

18. What is 'bias' in statistical studies?

    Systematic error leading to incorrect conclusions. Can occur in sampling, measurement, or analysis.

19. What is the purpose of a placebo?

    To account for the placebo effect in experiments. Controls for psychological effects of receiving treatment.

20. What is a 'matched pairs design'?

    Pairs of similar subjects receive different treatments for comparison. Controls for individual differences between subjects.

21. Which test compares proportions?

    Chi-square test. Tests independence or goodness of fit for categorical data.

22. Name the test for examining relationships between categorical variables.

    Chi-square test of independence. Determines if two categorical variables are independent.

23. What is the purpose of random assignment in experiments?

    To eliminate bias by evenly distributing confounding variables. Ensures treatment and control groups are comparable on average.

24. What is the purpose of replication in experiments?

    To confirm results are consistent and not due to random chance. Increases confidence in findings through repeated validation.

25. Calculate the margin of error: $SE = 2.5$, $z^* = 1.96$.

    4.9. Margin of error = $SE \times z^* = 2.5 \times 1.96$.

26. Which design reduces variability by pairing similar subjects?

    Matched pairs design. Effective method for controlling individual differences.

27. What does 'blocking' refer to in experiments?

    Grouping similar experimental units to reduce variability. Improves precision by controlling for known sources of variation.

28. What is a 'p-value'?

    Probability of observing data at least as extreme under $H_0$. Measures evidence against the null hypothesis.

29. Identify the null hypothesis symbol.

    $H_0$. Standard notation for the null hypothesis in statistics.

30. Identify the non-parametric alternative to the $t$-test.

    Wilcoxon signed-rank test. Does not assume normal distribution of data.

31. What is the primary goal of randomization?

    To reduce bias and confounding in the experiment. Creates comparable groups and eliminates systematic differences.

32. Define 'systematic sampling'.

    Selecting every nth member from a list after a random start. Simple method but may introduce patterns or bias.

33. What does 'internal validity' refer to?

    Extent to which cause-and-effect conclusions are warranted. Focuses on the accuracy of causal inferences.

34. What is 'confounding'?

    Distortion of the effect of one variable due to another. Makes it difficult to isolate the true effect.

35. What does 'statistical significance' mean?

    A result unlikely to occur by chance alone, below a threshold $p$-value. Indicates the result is not likely due to random variation.

36. Identify the alternative hypothesis symbol.

    $H_a$. Standard notation for the alternative hypothesis in statistics.

37. Find the critical value for a 95% confidence level (two-tailed).

    $z^*$ = 1.96. Standard normal value capturing 95% of the distribution.

38. What is a 'double-blind' experiment?

    Neither participants nor researchers know who receives treatment. Prevents bias from expectations affecting results.

39. What does 'ANOVA' stand for?

    Analysis of Variance. Statistical method comparing means across multiple groups.

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