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Mathematical Methods
Statistical inference
Chapter 14
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RESTS ON
Convert a percentage to a decimal
FOUNDATIONS
Percent as parts per hundred
FOUNDATIONS
For equiprobable outcomes, calculate an event probability as favourable outcomes divided by all outcomes and verify that it lies from zero to one
CH 12
Model an unordered selection with exact combination coefficients and combine disjoint composition cases by the sum rule
FOUNDATIONS
Evaluate or simplify a bounded factorial expression by expanding only the necessary consecutive factors and using the defining value at the zero index
FOUNDATIONS
Interpret the event after the conditioning event is known and restrict the denominator to that event's outcomes or table margin
CH 10
Calculate probabilities of specified success/failure patterns across independent trials, including all, only and at-least-one events, using complements where efficient
CH 11
Calculate the mean
CH 10
Signed numbers
CH 3
Apply the 68–95–99.7% empirical rule
CH 13
Multiplication and division
CH 9
Solve a one-step linear equation
CH 7
Identify a, b, c in zero form and calculate b squared minus 4ac with signs intact
CH 1
Solve equations with unknowns on both sides by collecting them onto one side
CH 1
Distinguish a random experiment, one outcome, the full sample space and an event as a subset of that sample space
CH 11
Populations, samples and sampling methods
Sample proportions and sampling distributions
Normal approximation for sample proportions
Confidence intervals for proportions
Sampling distribution simulations
Distinguish the population about which an inference is made from the sample that is actually measured; identify and state the population size N and sample size n, including cases where the population size is unknown or the sample is a census
Classify a stated numerical summary as a population parameter or a sample statistic; recognise that a sample statistic is measured from the sample and a population parameter is usually estimated, except when a census makes it known exactly
Recognise simple random, systematic, stratified and self-selected samples from a described selection procedure; judge whether the procedure gives every population member a known or equal selection probability and whether the sample is likely to be representative
Allocate a stratified sample in proportion to subgroup sizes: form the exact subgroup fraction of the total population, multiply by the target sample size, round individual allocations to whole members, and adjust so the allocations sum exactly to the target
Percentage of a quantity
Convert a fraction to a percentage
Multiply and divide fractions
Calculate a sample proportion as successes divided by sample size (p-hat = x/n) from raw counts, and reverse the relation to recover a count from a stated proportion and sample size; express the result as a reduced fraction, terminating decimal or percentage as the question requires
List all possible values of a sample proportion for a stated sample size by listing the possible success counts and dividing each by the sample size, including the boundary values zero and one
Construct the sampling distribution of a sample proportion when sampling without replacement from a finite binary population: count the samples giving each possible success count by exact combination products, divide by the total number of samples of that size, and verify that the probabilities sum exactly to one
Calculate an event probability or conditional probability for a sample proportion from its finite sampling distribution, using exact sums of the appropriate table entries
Calculate an exact probability as favourable counts over total counts using the same ordered or unordered model in numerator and denominator
Compute a combination coefficient exactly from its factorial ratio, cancelling before multiplication and respecting the selection constraints
Calculate an exact conditional probability as the intersection probability divided by the conditioning-event probability, with a nonzero conditioning event
Complete and validate a finite probability assignment: every probability is non-negative and the total is exactly one; use stated unequal outcome masses rather than assuming equiprobability
Calculate the exact probability of a specified number of successes in repeated independent Bernoulli trials using a combination count and complementary success/failure powers; combine disjoint terms for cumulative asks
Simplify a fraction
Test the chapter-adopted large-sample predicate for a stated population size, sample size and population proportion; where it fails, find the smallest sample size that satisfies it
State and use the mean (expected value) of the sample-proportion distribution as the population proportion p, including the linear derivation from the binomial count model
Calculate the variance p(1-p)/n and standard deviation of the sample-proportion distribution under the large-population binomial model, and interpret the standard deviation as the standard error of the sample proportion
Use the normal approximation for the sample proportion: standardise a sample-proportion boundary with the population mean and standard error, choose the correct tail or central region, and compute the required probability from the standard-normal percentage model
Recover an unknown population proportion or sample size from stated mean and standard-deviation (or standard-error) information about the sample proportion, including selecting the correct complementary root of the resulting quadratic equation from a stated direction
Calculate z = (x − mean) / standard deviation, retain its sign, and interpret its magnitude and direction as a distance from the mean in standard-deviation units
Use integer standard-score boundaries from −3 to 3 with the normal model to estimate central, interval and tail percentages or counts
Calculate the standard deviation
Solve two-step linear equations by inverse operations in reverse order
Substitute exact coefficients into the quadratic formula, keep the whole denominator 2a, take both sign branches and simplify
Select the conventional two-sided normal quantile for a stated confidence level and translate between confidence level, total tail area and one-tail area
Construct an approximate confidence interval for a population proportion from a sample count and sample size: estimate the standard error with p-hat, multiply by the level's quantile, and subtract/add that margin to p-hat; compute and interpret the margin of error
Find the sample size required to achieve a stated margin of error or interval half-width at a stated confidence level for a given or assumed sample proportion, and round up to a whole number of members
Recover the confidence level from a sample proportion, sample size and interval half-width (or symmetric endpoints) by forming the exact ratio of the half-width to the estimated standard error and reading the level from the conventional quantile table
Interpret an approximate confidence interval in context, including the confidence-in-the-method meaning, the distinction between a population parameter and a sample estimate, and the effect of a larger sample on the margin of error
Isolate a nominated subject in a linear literal equation by applying equivalent operations to the whole of both sides and simplifying only valid common factors
Read and interpret a synthetic simulated sampling distribution of sample proportions: identify its centre, spread and approximate shape, compare runs with different sample sizes or population proportions, and interpret the variation in confidence intervals between simulated samples
Choose a random device, outcome mapping, replacement rule, repetition unit and success condition that simulate the stated experiment without changing its probabilities
Calculate an estimated probability as the exact success count over trial count and interpret repeated-run variation without claiming that a finite simulation equals the theoretical probability