Applied Statistics

Units

1. 1

   Describing data

   2 lessons

   1. Graphs of dataDraw and read a histogram, a boxplot and a scatterplot, choosing bin widths and naming shape, centre, spread and outliers.
   2. Numerical summariesCompute and interpret the mean, median, standard deviation and interquartile range, and choose the summaries that resist outliers for skewed data.
2. 2

   Probability models for data

   3 lessons

   1. The binomial modelRecognize binomial counts, compute binomial probabilities, and use the mean np and standard deviation √(np(1 − p)) to judge whether a count is unusual.
   2. The Poisson modelModel counts of rare events in an interval with P(X = k) = e^{−λ}λᵏ/k!, check the fit against data, and rescale λ to a new interval.
   3. The normal model and normal quantile plotsFind normal probabilities and percentiles with z-scores, and judge from a normal quantile plot whether a normal model fits a data set.
3. 3

   Sampling distributions

   2 lessons

   1. The central limit theorem in practiceSimulated sample means from a skewed population pile into a bell narrower by √n; the standard error σ/√n, and how large n must be for skewed data.
   2. The t distributionReplacing σ by s gives the t statistic; the t curve's heavier tails beside the normal for 4 and 30 degrees of freedom, and t critical values.
4. 4

   Confidence intervals

   3 lessons

   1. Confidence intervals for a meanx̄ ± t* s/√n for the mean battery life of 25 batteries, what 95 percent confidence means (20 simulated intervals, about one missing the true mean), and the conditions.
   2. Confidence intervals for a proportionp̂ ± z*√(p̂(1 − p̂)/n) for a poll, the margin of error, and the plus-four interval when counts are small.
   3. Choosing a sample sizeSolving the margin-of-error formula for n, for a mean with a guessed σ and for a proportion with p = 0.5, and the cost of halving the margin.
5. 5

   Hypothesis tests

   3 lessons

   1. Testing a meanThe one-sample t test of whether bottles hold 500 mL: hypotheses, t statistic, the p-value as a tail area under the t curve, and a conclusion in context.
   2. Errors and powerType I and type II errors, the significance level, power as the area of the alternative's curve past the cut-off, and how n and effect size raise it, drawn with two overlapping curves.
   3. Comparing two meansWelch's two-sample t test for two teaching methods, the paired t test for before-and-after data, side-by-side boxplots and why pairing removes person-to-person spread.
6. 6

   Categorical data

   2 lessons

   1. Comparing two proportionsThe two-proportion z test and interval for a vaccine trial, the pooled proportion, and a segmented bar chart of the outcome by group.
   2. Chi-square testsGoodness of fit for a die and independence in a two-way table of preference by age group, expected counts, the chi-square statistic and its right-tail p-value.
7. 7

   Comparing several groups

   1 lesson

   1. One-way analysis of varianceThree diets' weight losses in side-by-side boxplots, between- and within-group variation, the F statistic and its p-value, and the conditions of equal spread and normality.
8. 8

   Regression and curve fitting

   4 lessons

   1. Fitting a line by least squaresThe least-squares line through a scatterplot of advertising and sales, slope b = r s_y/s_x, residuals as vertical gaps, and r² as the share of variation explained.
   2. Inference for regressionThe model y = β₀ + β₁x + ε, the standard error of the slope, the t test and interval for β₁, and confidence versus prediction bands around the line.
   3. Regression diagnosticsResidual plots that reveal curvature and fanning, a normal quantile plot of residuals, and outliers and high-leverage points that move the line.
   4. Fitting curves by transformationExponential growth data straightened by taking logarithms, a power law straightened on log-log axes, fitting the line and transforming back, with the curved fit over the data.
9. 9

   Resampling methods

   2 lessons

   1. Bootstrap confidence intervalsResampling the sample with replacement to see an estimate's variability, the bootstrap distribution of a median as a histogram, and the percentile interval.
   2. Permutation testsShuffling group labels to build the null distribution of a difference in means, the observed difference against it, and the p-value as the share at least as extreme.