StatisticsQ&A by dot point

Topic 1.9 Comparing Distributions of a Quantitative Variable: compare two or more distributions of a quantitative variable by shape, center, spread, and unusual features, in context, using comparative language.

Topic 1.6 Describing the Distribution of a Quantitative Variable: describe a quantitative distribution by its shape, center, spread, and unusual features (outliers, gaps, clusters) in context.

Topic 1.8 Graphical Representations of Summary Statistics: construct and interpret boxplots from the five-number summary, and identify outliers using the 1.5 times IQR rule.

Topic 1.1 Introducing Statistics - What Can We Learn from Data?: identify questions to be answered, based on variation in one-variable data, and recognize what a data set can and cannot tell us.

Topic 1.4 Representing a Categorical Variable with Graphs: choose, construct, and interpret bar graphs and other displays of a single categorical variable, and describe the distribution of categories.

Topic 1.3 Representing a Categorical Variable with Tables: build and interpret frequency and relative frequency tables for a single categorical variable, and read proportions and percentages from them.

Topic 1.5 Representing a Quantitative Variable with Graphs: construct and interpret dotplots, stem-and-leaf plots, and histograms for a quantitative variable, and choose an appropriate display.

Topic 1.7 Summary Statistics for a Quantitative Variable: calculate and interpret measures of center (mean, median) and spread (range, IQR, standard deviation, variance), and judge their resistance to outliers.

Topic 1.2 The Language of Variation - Variables: classify variables as categorical or quantitative, and quantitative variables as discrete or continuous, and explain why the type determines the appropriate graphs and statistics.

Topic 1.10 The Normal Distribution: use z-scores, the empirical (68-95-99.7) rule, and the standard normal model to find proportions and percentiles for approximately normal data.

Topic 2.9 Analyzing Departures from Linearity: identify outliers, high-leverage, and influential points in regression, and use transformations to model a non-linear relationship.

Topic 2.5 Correlation: calculate and interpret the correlation coefficient r, understand its properties (range, unit-free, resistance), and recognize what it can and cannot tell you.

Topic 2.1 Introducing Statistics - Are Variables Related?: identify questions about the association between two variables, distinguish association from causation, and recognize what two-variable data can answer.

Topic 2.8 Least Squares Regression: determine the least-squares regression line from summary statistics, and interpret the coefficient of determination r-squared and the standard deviation of the residuals.

Topic 2.6 Linear Regression Models: write, interpret, and use a least-squares regression equation to predict a response, interpreting the slope and intercept in context, and recognizing the danger of extrapolation.

Topic 2.4 Representing the Relationship Between Two Quantitative Variables: construct and describe scatterplots by direction, form, strength, and unusual features, in context.

Topic 2.2 Representing Two Categorical Variables: construct and interpret two-way (contingency) tables and segmented or side-by-side bar graphs for two categorical variables.

Topic 2.7 Residuals: calculate and interpret residuals, construct and read residual plots, and use them to assess whether a linear model is appropriate.

Topic 2.3 Statistics for Two Categorical Variables: calculate joint, marginal, and conditional relative frequencies from a two-way table, and use conditional distributions to judge association.

Topic 3.7 Inference and Experiments: use the presence or absence of random selection and random assignment to determine the scope of inference, that is, whether results generalize to a population and whether a causal conclusion is justified.

Topic 3.1 Introducing Statistics: Do the Data We Collected Tell the Truth? Recognize that the method of data collection determines the kinds of conclusions that can be drawn, and that poorly collected data cannot be fixed by analysis.

Topic 3.5 Introduction to Experimental Design: identify the components of an experiment (units, treatments, response) and apply the principles of comparison, random assignment, replication, and control, including blinding and the placebo effect.

Topic 3.2 Introduction to Planning a Study: distinguish observational studies from experiments, identify explanatory and response variables, and recognize that only an experiment with imposed treatments can support a causal conclusion.

Topic 3.4 Potential Problems with Sampling: identify undercoverage, voluntary response, convenience, nonresponse, and response bias, explain how each distorts results, and recognize that bias is not reduced by a larger sample.

Topic 3.3 Random Sampling and Data Collection: describe and distinguish simple random, stratified, cluster, and systematic random sampling, and explain why random selection supports generalization to a population.

Topic 3.6 Selecting an Experimental Design: compare completely randomised, randomised block, and matched pairs designs, and explain how blocking and pairing control a known source of variation to make treatment effects clearer.

Topic 4.9 Combining Random Variables: apply the rules for the mean and variance of a linear transformation and of sums and differences of random variables, adding variances (not standard deviations) for independent variables.

Topic 4.5 Conditional Probability: calculate and interpret conditional probabilities using the definition and the multiplication rule, including from two-way tables and tree diagrams.

Topic 4.2 Estimating Probabilities Using Simulation: design and carry out a simulation using a chance device or random numbers to estimate a probability as a long-run relative frequency.

Topic 4.6 Independent Events and Unions of Events: determine whether events are independent, apply the multiplication rule for independent events, and combine the addition and multiplication rules to find probabilities of unions and intersections.

Topic 4.1 Introducing Statistics: Random and Non-Random Patterns? Recognize that random processes produce patterns, and that probability provides the framework for deciding whether an observed pattern is surprising or consistent with chance.

Topic 4.3 Introduction to Probability: apply the basic properties of probability (range, total of one, complement rule) and the law of large numbers to compute and interpret probabilities of events.

Topic 4.7 Introduction to Random Variables and Probability Distributions: define discrete random variables, represent and interpret their probability distributions, and use them to find probabilities of events.

Topic 4.10 Introduction to the Binomial Distribution: identify binomial settings (BINS conditions) and use the binomial probability formula to find the probability of a given number of successes in a fixed number of trials.

Topic 4.8 Mean and Standard Deviation of Random Variables: calculate and interpret the mean (expected value), variance, and standard deviation of a discrete random variable from its probability distribution.

Topic 4.4 Mutually Exclusive Events: identify mutually exclusive (disjoint) events and apply the addition rule, including the general addition rule that subtracts the overlap, to find the probability of a union.

Topic 4.11 Parameters for a Binomial Distribution: calculate and interpret the mean and standard deviation of a binomial random variable using the shortcut formulas, and describe how the distribution's shape depends on n and p.

Topic 4.12 The Geometric Distribution: identify a geometric setting (waiting for the first success), compute geometric probabilities, and find the mean of a geometric random variable.

Topic 5.4 Biased and Unbiased Point Estimates: define an unbiased estimator (its sampling distribution centers on the parameter), and distinguish the bias of an estimator from its variability.

Topic 5.1 Introducing Statistics: Why Is My Sample Not Like Yours? Recognize sampling variability, the difference between a parameter and a statistic, and that a statistic varies from sample to sample in a predictable way.

Topic 5.8 Sampling Distributions for Differences in Sample Means: describe the mean, standard deviation, and shape of the sampling distribution of the difference between two independent sample means, adding variances and checking the conditions for normality.

Topic 5.6 Sampling Distributions for Differences in Sample Proportions: describe the mean, standard deviation, and shape of the sampling distribution of the difference between two independent sample proportions, and check the conditions for the normal model.

Topic 5.7 Sampling Distributions for Sample Means: describe the mean, standard deviation, and shape of the sampling distribution of a sample mean, using the central limit theorem and the standard deviation formula sigma over root n.

Topic 5.5 Sampling Distributions for Sample Proportions: describe the mean, standard deviation, and shape of the sampling distribution of a sample proportion, and check the conditions (10% and large counts) for the normal model.

Topic 5.3 The Central Limit Theorem: state and apply the central limit theorem, that the sampling distribution of the sample mean becomes approximately normal as the sample size grows, regardless of the population's shape.

Topic 5.2 The Normal Distribution, Revisited: revisit the normal model and z-scores in the context of distributions of statistics, finding proportions and using the standard normal as the basis for later inference.

Topic 6.11 Carrying Out a Test for the Difference of Two Population Proportions: compute the two-sample z test statistic using the pooled standard error, find the P-value, and state a conclusion in context.

Topic 6.6 Concluding a Test for a Population Proportion: compute the standardized z test statistic and P-value for a one-sample proportion test, compare to the significance level, and state a conclusion in context.

Topic 6.8 Confidence Intervals for the Difference of Two Proportions: check the conditions and construct a two-sample z-interval for the difference between two population proportions, using the unpooled standard error.

Topic 6.2 Constructing a Confidence Interval for a Population Proportion: identify the conditions, compute the point estimate, critical value, standard error, and margin of error, and construct and interpret a one-sample z-interval for a proportion.

Topic 6.5 Interpreting P-Values: define the P-value as the probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed, and interpret it in context.

Topic 6.1 Introducing Statistics: Why Be Normal?: explain how the approximately normal sampling distribution of a sample proportion lets us quantify uncertainty and make inferences about an unknown population proportion.

Topic 6.9 Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions: use a two-sample proportion interval to judge whether a difference exists and to evaluate claims about the size and direction of that difference.

Topic 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion: use a confidence interval for a proportion to evaluate whether a claimed value is plausible, and discuss the effect of confidence level and sample size on the interval.

Topic 6.7 Potential Errors When Performing Tests: distinguish Type I and Type II errors and their consequences, define the power of a test, and explain how significance level, sample size, and effect size affect error probabilities and power.

Topic 6.4 Setting Up a Test for a Population Proportion: state null and alternative hypotheses about a population proportion, identify the significance level, and verify the conditions for a one-sample z-test.

Topic 6.10 Setting Up a Test for the Difference of Two Population Proportions: state the hypotheses about the difference of two proportions, identify the significance level, and verify the conditions for a two-sample z-test using the pooled proportion.

Topic 7.5 Carrying Out a Test for a Population Mean: compute the t test statistic with n minus 1 degrees of freedom, find the P-value, compare to the significance level, and state a conclusion in context.

Topic 7.9 Carrying Out a Test for the Difference of Two Population Means: compute the two-sample (or paired) t test statistic, find the P-value, compare to the significance level, and state a conclusion in context.

Topic 7.6 Confidence Intervals for the Difference of Two Means: check the conditions and construct a two-sample t-interval for the difference between two population means, including the paired case, using the unpooled standard error.

Topic 7.2 Constructing a Confidence Interval for a Population Mean: check the conditions and construct a one-sample t-interval for a population mean, using the t critical value, the standard error, and the correct degrees of freedom.

Topic 7.1 Introducing Statistics: Should I Worry About Error?: explain why a sample mean varies from sample to sample, why this sampling variability creates uncertainty about the population mean, and how inference quantifies that error.

Topic 7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval: use a one-sample mean interval to judge whether a claimed mean is plausible, and explain how confidence level and sample size affect the interval.

Topic 7.7 Justifying a Claim About the Difference of Two Means Based on a Confidence Interval: use a two-sample (or paired) mean interval to judge whether the means differ and to assess claims about the size and direction of the difference.

Topic 7.10 Skills Focus: Selecting, Implementing, and Communicating Inference Procedures: identify the appropriate confidence interval or significance test for a scenario (proportion or mean, one or two samples, paired or independent), and carry it out and communicate the result correctly.

Topic 7.4 Setting Up a Test for a Population Mean: state the null and alternative hypotheses about a population mean, identify the significance level, and verify the conditions for a one-sample t-test.

Topic 7.8 Setting Up a Test for the Difference of Two Population Means: state the hypotheses about the difference of two means, decide between a two-sample and a paired procedure, identify the significance level, and check the conditions.

Topic 8.3 Carrying Out a Chi-Square Test for Goodness of Fit: compute the chi-square statistic from observed and expected counts, find the P-value using k minus 1 degrees of freedom, and state a conclusion in context.

Topic 8.6 Carrying Out a Chi-Square Test for Homogeneity or Independence: compute the chi-square statistic from a two-way table, find the P-value using (rows minus 1)(columns minus 1) degrees of freedom, and state a conclusion in context.

Topic 8.4 Expected Counts in Two-Way Tables: compute the expected count for each cell of a two-way table under the null hypothesis using the row total times column total divided by the grand total.

Topic 8.1 Introducing Statistics: Are My Results Unexpected?: explain why comparing observed counts across several categories to expected counts motivates the chi-square family of tests.

Topic 8.7 Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data: choose among the one-proportion, two-proportion, and chi-square (goodness of fit, homogeneity, independence) procedures based on the scenario.

Topic 8.2 Setting Up a Chi-Square Goodness of Fit Test: state the hypotheses for a goodness-of-fit test, compute expected counts from a claimed distribution, and verify the conditions.

Topic 8.5 Setting Up a Chi-Square Test for Homogeneity or Independence: distinguish a test of homogeneity from a test of independence based on the design, state the appropriate hypotheses, and check the conditions.

Topic 9.5 Carrying Out a Test for the Slope of a Regression Model: compute the t test statistic for the slope using the standard error, find the P-value with n minus 2 degrees of freedom, and state a conclusion in context.

Topic 9.2 Confidence Intervals for the Slope of a Regression Model: check the regression conditions and construct a t-interval for the population slope using the sample slope, its standard error, and n minus 2 degrees of freedom.

Topic 9.1 Introducing Statistics: Do Those Points Align?: explain why a sample regression slope varies from sample to sample, motivating inference about the true population slope of a linear model.

Topic 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval: use a slope interval to judge whether a linear relationship exists and to evaluate claims about the size and direction of the slope.

Topic 9.6 Skills Focus: Selecting an Appropriate Inference Procedure: choose the correct inference procedure (proportion, mean, chi-square, or slope; interval or test; one or two samples; paired or independent) for any scenario across the whole course.

Topic 9.4 Setting Up a Test for the Slope of a Regression Model: state the null and alternative hypotheses about the population slope, identify the significance level, and verify the regression conditions for a t-test.