Course Standards

I can classify any variable, identify study components (population, sample, parameter, statistic) with proper notation, name the sampling method used, and explain how its design could bias results.

Classify any variable by type and sub-type; identify the population, sample, parameter, and statistic in a research scenario using proper notation (\(\mu, \bar{x}, \sigma, s, p, \hat{p}\)); identify the sampling method used in a study; and evaluate whether the design is likely to produce a representative sample, explaining the source and likely direction of any bias.

I can pick the right graph for a variable, build and read frequency tables, and spot a misleading graph.

Select the appropriate graph type for a given variable and research question; construct and read frequency distributions correctly; identify misrepresentation in a graph and explain what it obscures.

I can compute the standard measures of center and spread, then choose and justify the right ones for a distribution.

Calculate the measures of center and spread — mean, median, mode, range, variance, standard deviation (\(n-1\) denominator for sample data), the five-number summary, IQR, and the coefficient of variation (\(CV = \frac{s}{\bar{x}} \times 100\%\)); then select and justify the appropriate measures for a given distribution.

I can read a real-world situation and calculate probabilities of simple and compound events, including when two events overlap or are mutually exclusive.

Set up and calculate probabilities of simple, complementary, and compound events directly from a described real-world situation — not only from a probability expression you are handed; and apply the addition rule correctly based on whether the events are mutually exclusive.

I can read a real-world situation, calculate conditional and joint probabilities, and determine whether two events are mutually exclusive, independent, both, or neither.

Compute conditional probabilities \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) with proper conditioning direction; apply the general multiplication rule \(P(A \cap B) = P(A)\,P(B|A)\) to find joint probabilities, with the independent special case \(P(A \cap B) = P(A)\,P(B)\); formally test for independence using both the conditional and multiplication definitions; and classify a pair of events as mutually exclusive, independent, both, or neither, justifying the relationship in writing.

I can read a real-world situation, count its possible outcomes — deciding when order matters — and use the count to find a probability.

Apply the Fundamental Counting Principle, arrangements (\(_nP_r\), ordered selection of \(r\) — including the all-elements permutation \(n!\)), and combinations (\(_nC_r\) or \(\binom{n}{r}\)) to count outcomes; determine whether order matters in context and select the correct formula; use counting results to compute probabilities over equally-likely sample spaces.

I can build a valid probability distribution for a random variable, compute and interpret its mean, variance, and standard deviation, and find cumulative probabilities.

Identify a random variable as discrete or continuous; construct a valid probability mass function (PMF) and verify that all probabilities sum to 1; calculate \(E(X)\), \(Var(X)\), and \(\sigma_X\); compute cumulative probabilities (\(P(X \le k)\), \(P(X > k)\), \(P(a < X \le b)\)); interpret \(E(X)\) as a long-run average — not the most likely outcome — and explain why \(E(X^2) \neq [E(X)]^2\) in general.

I can check the BINS conditions and compute binomial probabilities (including "at least"/"at most").

Verify all four BINS conditions before applying the binomial model, and recognize when it does not apply; calculate exact and cumulative binomial probabilities; translate "at most \(k\)," "at least \(k\)," "more than \(k\)," and "fewer than \(k\)" correctly into probability expressions.

I can use the normal distribution model to find probabilities from values (standardizing to z first) and find values from probabilities (inverse normal).

Standardize a general normal variable \(X \sim N(\mu, \sigma^2)\) to \(Z\) and compute probabilities \(P(Z < a)\), \(P(Z > a)\), and \(P(a < Z < b)\) using the \(z\)-table; and perform inverse normal lookups to find the value \(x\) corresponding to a given percentile or area, unstandardizing with \(x = \mu + z^* \sigma\).

I can read a probability problem with no method named, recognize which approach it needs, and compute the answer.

Given a probability scenario with no method or section specified, recognize which approach applies — direct counting / the Fundamental Counting Principle, the addition or multiplication rules, conditional probability, or a named distribution (binomial or normal); justify the choice from the structure of the problem; and compute the requested probability.

I can describe the sampling distribution of \(\bar{x}\) and calculate probabilities for sample means, and explain how the Central Limit Theorem applies.

Using the standard error \(\sigma/\sqrt{n}\), calculate probabilities for sample means; explain what the Central Limit Theorem states about the shape of the sampling distribution of \(\bar{x}\) and when it applies; and distinguish the distribution of sample means from individual observations.

I can construct and interpret a confidence interval for a population mean (\(z\) vs. \(t\)) or proportion (\(z\)), check the right conditions, interpret the confidence level, and solve for required sample sizes.

Construct a confidence interval as point estimate \(\pm\) margin of error. For a mean: use \(z\) when \(\sigma\) is known or \(n \ge 30\), and \(t\) when \(\sigma\) is unknown and \(n < 30\) (\(df = n-1\)). For a proportion: always use \(z\), with point estimate \(\hat{p}\) and conditions \(n\hat{p} \ge 10\) and \(n(1-\hat{p}) \ge 10\). Check conditions; interpret the interval in context without attaching probability to the fixed parameter; explain what the confidence level means and how width responds to \(n\), variability, and confidence; and determine the sample size needed for a target margin of error (using a prior \(\hat{p}\) or \(p^* = 0.5\) worst case for a proportion).

I can run a five-step hypothesis test for a population mean (\(z\) vs. \(t\)) or proportion (\(p_0\) in the standard error), define a \(p\)-value, interpret a non-significant result, and explain Type I and Type II errors.

Execute all five steps of a hypothesis test for a population mean or proportion. For a mean: use \(z\) when \(\sigma\) is known or \(n \ge 30\), and \(t\) when \(\sigma\) is unknown and \(n < 30\). For a proportion: run a one-proportion \(z\)-test, checking \(np_0 \ge 10\) and \(n(1-p_0) \ge 10\) and building the standard error from the null value \(p_0\) (not \(\hat{p}\)). Check conditions; compute the test statistic and find or bound the \(p\)-value; state the conclusion in context; define \(p\)-value correctly as a conditional probability; interpret "fail to reject" correctly; and identify Type I and Type II errors in context.

I can read an unstructured scenario, decide which inference procedure it calls for, carry it out, and answer the original question.

Given a scenario with no procedure named, decide whether it calls for a confidence interval or a hypothesis test, and whether the parameter is a mean or a proportion; select \(z\) vs. \(t\) and check the relevant conditions; carry out the chosen procedure; and state the answer in the context of the original question.

I can describe scatter plots, calculate and interpret correlation (\(r\)) and the coefficient of determination (\(r^2\)), check Pearson conditions, and explain why correlation is not causation.

Calculate and interpret Pearson \(r\) and \(r^2\); describe scatter plots by direction, form, strength, and outliers; check conditions for Pearson \(r\); and explain why correlation does not imply causation.

I can find the least-squares regression line, interpret its slope and intercept, use it to make predictions (distinguishing interpolation from extrapolation), and identify outliers and high-leverage points.

Calculate the least-squares regression line using \(b = r(s_y/s_x)\) and \(a = \bar{y} - b\bar{x}\) (and verify it passes through \((\bar{x}, \bar{y})\)); interpret slope and intercept in context; use the line for prediction, distinguishing interpolation from extrapolation and explaining the risk of extrapolation; and identify regression outliers and influential/high-leverage points and describe their effects on the line.

I can run a chi-square test of independence and interpret it as association (not causation).

Calculate expected frequencies using \(E = (\text{row total} \times \text{column total})/\text{grand total}\); compute \(\chi^2 = \sum \frac{(O-E)^2}{E}\); execute the five-step chi-square test with \(df = (r-1)(c-1)\); verify that all expected frequencies are \(\ge 5\); interpret the result as evidence of association, not causation.

I can look at two variables and a research question and decide whether it calls for a chi-square test or a correlation/regression analysis — then carry it out.

Given two variables and a research question, determine the analysis from the variable types: a chi-square test of independence for two qualitative variables, or correlation and least-squares regression for two quantitative variables; justify the choice; carry out the appropriate analysis; and state the conclusion as association (never causation) in context.