Introductory Statistics

Units

1. 1

   Collecting data

   7 lessons

   1. Statistical studies, investigative questions and variablesIdentify study components, classify variables and summaries, and write an investigative question.
   2. Planning a study: census, observational study or experiment, and the investigative questionClassify a study, write a question its design can address, and justify the scope of its conclusions.
   3. Random sampling methodsDescribe four random sampling methods and justify a method for a school survey.
   4. Potential problems with sampling: biasIdentify a possible bias, explain who is affected, and state its likely direction when evidence supports one.
   5. Principles of experimental designWrite and justify an experiment with a comparison group, random assignment, replication, controlled conditions, a measured response and appropriate conclusions.
   6. Completely randomised, randomised block and matched pairs designsChoose, describe and justify a design for comparing two treatments.
   7. Inference and experiments: scope of conclusions and statistical significanceUse a shuffle result and a study’s design to write a justified conclusion.
2. 2

   Describing data

   6 lessons

   1. Tables and graphs for one categorical variableConstruct categorical tables and graphs, compare group rates, and justify claims with proportions in context.
   2. Dotplots, stemplots and histogramsConstruct three displays of quantitative data and explain what each shows or hides.
   3. Describing a quantitative distribution: shape, center, variability, unusual featuresDescribe a quantitative distribution’s shape, center, variability, and unusual features in context, using evidence from its graph.
   4. Summary statistics: center, position, variability and outliersCalculate commute summaries, identify outliers, and justify statistics and units for a skewed distribution.
   5. Five-number summary and boxplotsConstruct a modified boxplot and interpret its quarters, outliers, and mean–median relationship.
   6. Comparing distributions and z-scoresCompare distributions in context and use z-scores to compare scores relative to their groups.
3. 3

   Probability

   5 lessons

   1. Random processes and estimating probabilities by simulationDesign a simulation and use its results to estimate an event’s probability.
   2. Probability, complements and mutually exclusive eventsCalculate probabilities by counting or using complements, and justify when events are mutually exclusive.
   3. Conditional probability and the general multiplication ruleCalculate conditional probabilities from trees and two-way tables, including when the question reverses the tree’s first branch.
   4. Independent events and unions of eventsCheck independence using probabilities and calculate “or” probabilities without counting an overlap twice.
   5. Two categorical variables: two-way tables, conditional distributions and associationCalculate shares from a two-way table and use within-group comparisons to judge association.
4. 4

   Random variables and distributions

   6 lessons

   1. Discrete random variables and probability distributionsConstruct a discrete probability distribution and use it to find probabilities described in words.
   2. Mean and standard deviation of a discrete random variableCalculate and interpret a discrete random variable’s mean and standard deviation.
   3. Transforming and combining random variablesCalculate means and SDs for transformed sums and differences, then find probabilities when the combined variable is normal.
   4. The binomial distributionJustify a binomial model; calculate and interpret its probabilities, mean, and standard deviation.
   5. The geometric distributionModel trials until a first success; calculate geometric probabilities, mean, and standard deviation.
   6. The normal distribution: empirical rule and normal calculationsCalculate and interpret normal-model probabilities and percentile cutoffs using a labelled curve.
5. 5

   Sampling distributions

   5 lessons

   1. Point estimators: bias and variabilityCompare point estimators using the centres and spreads of their sampling distributions.
   2. Sampling variability and sampling distributions (simulation)Tell a population, a sample and a sampling distribution apart, describe a simulated sampling distribution by centre, spread and shape, and use one to judge how unusual a sample result is.
   3. The central limit theorem and probabilities about x-barUse the central limit theorem to say when the sampling distribution of x̄ is approximately normal, give its centre and SD, justify the conditions, and find a probability about a sample mean.
   4. The sampling distribution of a sample proportionDescribe its centre, spread and shape, then calculate and interpret a probability.
   5. The sampling distribution of a sample meanJustify with numbers a normal model for x̄, interpret probabilities about x̄ as statements about samples, and use the standard error when σ is unknown.
6. 6

   Inference for proportions

   9 lessons

   1. Constructing a confidence interval for a population proportionConstruct a one-sample z-interval for a population proportion, verify its conditions, and plan a sample size.
   2. Interpreting a confidence interval and the confidence levelInterpret a proportion interval and its confidence level, assess a claim, and explain changes in width.
   3. Setting up a significance test for a population proportionDefine the population proportion, write hypotheses from the question, name the test, and check its conditions.
   4. Interpreting p-valuesInterpret a p-value in context, identifying the null assumption and the results counted as more extreme.
   5. Carrying out and concluding a test for a population proportionCalculate a one-proportion z-test and state a justified conclusion about the population.
   6. Type I and Type II errors and powerIdentify both errors in context, calculate power, and explain how testing choices affect mistakes.
   7. The sampling distribution of a difference in sample proportionsDescribe its centre, spread and shape; check conditions; calculate and interpret probabilities.
   8. Confidence intervals for a difference in population proportionsConstruct and interpret a two-sample z-interval for a difference in population proportions.
   9. Significance tests for a difference in population proportionsSet up a two-proportion z-test, check its conditions, and conclude with appropriate scope.
7. 7

   Inference for means

   8 lessons

   1. t-distributions and the confidence interval for a population meanConstruct, justify, and interpret a one-sample t-interval for a population mean.
   2. Justifying claims with a confidence interval for a meanInterpret a t-interval for a population mean and use it to evaluate a claimed mean.
   3. Significance test for a population meanCarry out and justify a one-sample t-test for a population mean.
   4. Inference for a mean difference: matched pairsIdentify paired data, check conditions, and test or estimate a population mean difference using one-sample t procedures.
   5. The sampling distribution of a difference in sample meansDescribe its centre, spread, and shape; check conditions; calculate a probability.
   6. Confidence intervals for a difference in population meansConstruct, interpret, and justify a confidence interval for the difference between two population means.
   7. Significance tests for a difference in population meansTest a difference between two independent population means and state a conclusion with the right scope.
   8. Selecting, implementing and communicating inference proceduresChoose an inference method, carry it out, and justify a conclusion in context.
8. 8

   Chi-square tests

   4 lessons

   1. Expected counts in two-way tablesCalculate expected counts from table margins and check the chi-square expected-count condition.
   2. Chi-square goodness-of-fit testTest whether a population’s colour shares match a claim, and state a conclusion in context.
   3. Setting up chi-square tests for homogeneity and independenceChoose the test from the study design, state hypotheses in context, and check conditions.
   4. Carrying out chi-square tests for homogeneity and independenceCalculate a chi-square statistic and p-value, then justify a contextual conclusion and choose the appropriate categorical test.
9. 9

   Correlation and regression

   7 lessons

   1. Scatterplots: form, direction, strength and unusual featuresPlot paired measurements and describe a scatterplot’s direction, form, strength and unusual features in context.
   2. CorrelationInterpret r in context and use a scatterplot to judge what it does and does not show.
   3. Linear regression models and residualsPredict a car’s price, interpret its residual, judge the residual plot, and recognize extrapolation.
   4. Least-squares regression: slope, intercept and r-squaredCalculate a least-squares line and interpret its slope, intercept, R-sq and S in context.
   5. Outliers, influential points and transformationsDistinguish outliers from influential points, assess a log transformation, and predict in original units.
   6. Sampling distribution of the slope and the confidence interval for betaConstruct and interpret a justified confidence interval for a population regression slope from computer output.
   7. Significance test for the slope of a regression lineTest whether a population regression slope is zero and interpret the result within the study’s design.