AP Statistics

What people get wrong

1. Dividing by n instead of n-1 when computing a sample standard deviation by hand. Instead, use $n-1$ (or trust the calculator's Sx, not σx, for sample data) since dividing by n underestimates spread.
2. Concluding causation from an observational study (e.g., 'vitamin use causes lower heart disease rates'). Instead, state only that the variables are associated and note a plausible confounding variable, since causal claims require random assignment.
3. Confusing stratified and cluster sampling. Instead, remember that stratified sampling takes a random sample from every group (each stratum is represented), while cluster sampling randomly selects a few whole groups and surveys everyone within them.
4. Reporting the mean and standard deviation for a skewed distribution or one with outliers. Instead, check the shape first (from a graph or the mean-vs-median comparison) and report the median and IQR when the data are not roughly symmetric.

What people get wrong

1. Writing P(A|B) when the problem gives P(B|A), or vice versa — conditional probability is not symmetric. Always identify which event is the 'given' condition before dividing.
2. Assuming two events are independent because they seem unrelated in context. Instead, compute P(A), P(B), and P(A and B) (or P(B|A)) and check the independence equation numerically.
3. Using the simple addition rule P(A)+P(B) for events that overlap. Instead, always subtract P(A and B) unless the problem explicitly states or a Venn diagram confirms the events are mutually exclusive.
4. Applying the binomial formula to a without-replacement scenario drawn from a small population, where trials are not actually independent. Instead, check that the population is much larger than the sample (a common rule of thumb is at least 10 times the sample size) before treating it as binomial.

What people get wrong

1. Using the pooled proportion when constructing a two-proportion confidence interval. The pooled proportion is only for the significance test of H0: p1 = p2; confidence intervals must use each sample's own $\hat p_1$ and $\hat p_2$ in the standard error.
2. Checking the Large Counts condition using n alone instead of the actual expected successes and failures. Always compute $n\hat p$, $n(1-\hat p)$ (or expected counts in a table) and confirm each is at least 10 for proportions or 5 for chi-square before proceeding.
3. Interpreting a confidence interval as 'there is a 95% probability the true proportion is in this interval.' The parameter is fixed, not random — the correct interpretation is about the long-run percentage of intervals from repeated sampling that would capture the true value.
4. Mixing up degrees of freedom formulas — using (r−1)(c−1) for a goodness-of-fit test or k−1 for a two-way table. Match the formula to the test: k−1 for goodness of fit, (r−1)(c−1) for independence or homogeneity.

What people get wrong

1. Using z* instead of t* whenever σ is unknown (which is almost always). Instead, always use the t-distribution with the correct degrees of freedom when estimating the standard deviation from the sample.
2. Treating two independent samples as if they should use a pooled-variance formula. AP Statistics convention is to always use the unpooled two-sample t procedure for means, not the pooled t-test taught in some other courses.
3. Confusing paired and independent designs — using the two-sample formula on data that are actually paired (same subjects measured twice) inflates the standard error and gives wrong conclusions. Instead, recognize paired designs and reduce to differences first.
4. Interpreting a confidence interval as 'there is a 95% probability the true mean is in this specific interval' or '95% of the data lies in this interval.' Instead, state that in repeated sampling, about 95% of intervals constructed this way would capture the true population mean.

What people get wrong

1. Interpreting r² as if it were r (e.g. saying 'r² = 0.72 means a strong correlation of 0.72'). Instead, always state r² as a percentage of variation explained, and square-root it (keeping the sign of the slope) if you actually need r.
2. Extrapolating the regression line beyond the range of observed x-values and treating the prediction as reliable. Instead, explicitly flag predictions outside the data range as extrapolation and note the relationship may not hold there.
3. Interpreting slope or intercept without units or context, e.g. 'y increases by 3.85' instead of 'exam score increases by 3.85 points, on average, for each additional hour studied.' The AP rubric requires context in every interpretation.
4. Running a t-test or CI for the slope without first checking the residual plot for linearity and constant spread. Instead, state explicitly that the residual plot shows no pattern and roughly constant spread before proceeding to inference.