NCEA Level 3 Statistics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Expected value of X

E(X)=Σx·P(x), the probability-weighted average of all possible values of a discrete random variable; it represents the long-run mean over many repetitions.

Variance formula

Var(X)=E(X²)−[E(X)]², where E(X²)=Σx²·P(x). The standard deviation is the positive square root of the variance and uses original units.

Binomial conditions

Fixed number of independent trials n, two outcomes per trial, and constant success probability p. Then E(X)=np and Var(X)=np(1−p).

Binomial probability formula

P(X=r)=C(n,r)·p^r·(1−p)^(n−r), where C(n,r) counts the number of orderings of r successes among n trials and p is the success probability.

Standard normal variable

z=(x−μ)/σ converts a normal variable to the standard normal with mean 0 and standard deviation 1, allowing probabilities to be read from tables or software.

Continuity correction

Adjusting a discrete value by 0.5 before using a continuous normal approximation, e.g. P(X≤k) becomes P(X<k+0.5), to account for the gap between discrete values.

Inverse normal

Given a probability, find z from the standard normal distribution, then convert back with x=μ+z·σ. For upper-tail areas use the complementary lower-tail area or negate z.

Sum of independent variables

For independent X and Y: E(aX+bY)=aE(X)+bE(Y) and Var(aX+bY)=a²Var(X)+b²Var(Y). Variances always add, even when the means are subtracted.

Standard error of mean

SE=σ/√n, the standard deviation of the sampling distribution of the sample mean. It decreases as sample size grows, proportional to 1/√n.

Confidence interval for mean

x̄ ± z·σ/√n for known σ. The interval estimates the population mean, and its width depends on confidence level, standard deviation and sample size.

Null and alternative hypotheses

H₀ states no effect or a specific parameter value; Hₐ states the effect being tested. They are written about population parameters, not sample statistics.

P-value meaning

The probability, assuming H₀ is true, of obtaining a result at least as extreme as observed in the direction of Hₐ. It is not the probability that H₀ is true.

Type I error

Rejecting a true null hypothesis. Its probability equals the significance level α when H₀ is true, so lowering α reduces the chance of this error.

Type II error

Failing to reject a false null hypothesis. Its probability depends on the true parameter value, sample size and α; larger samples reduce it.

Paired data test

Each unit gives two linked measurements; the differences are analysed as one sample, testing H₀: μ_d=0 using the mean and standard deviation of the differences.

Chi-square goodness of fit

Tests whether observed category counts match expected counts from a hypothesised distribution, using χ²=Σ(O−E)²/E with df = categories − 1.

Chi-square independence test

Tests association between two categorical variables in an r×c table. Expected count is (row total×column total)/grand total and df=(r−1)(c−1).

Correlation coefficient r

A number between −1 and 1 measuring the strength and direction of linear association between two quantitative variables. It ignores nonlinear patterns and is unaffected by variable order.

Coefficient of determination

r², the proportion of variation in the response variable explained by its linear regression on the explanatory variable. It ranges from 0 to 1.

Least-squares regression line

ŷ=b₀+b₁x with b₁=r·(s_y/s_x) and b₀=ȳ−b₁x̄. It minimises the sum of squared residuals and passes through (x̄, ȳ).

Interpreting regression slope

The predicted change in the response variable for each one-unit increase in the explanatory variable, with units of response per unit of explanatory variable.

Residual

The difference between an observed value and the value predicted by the regression line, y−ŷ. Random scatter of residuals supports a linear model.

Extrapolation

Using a regression model to predict outside the observed range of the explanatory variable. Such predictions are unreliable because the relationship may change.

Influential point

An observation whose removal substantially changes the regression line, slope or correlation. Its effect should be identified and justified, not silently deleted.

Sampling bias

Systematic error from a sample that is not representative of the population, such as a voluntary-response or convenience sample. Larger samples do not remove it.

Central limit theorem

For sufficiently large n, the sample mean is approximately normally distributed with mean μ and standard error σ/√n, whatever the shape of the population distribution.

Stratified sampling

The population is divided into strata that differ in a relevant characteristic, then random samples are taken from each stratum, often in proportion to stratum size.

Conditional probability

P(A|B)=P(A∩B)/P(B), the probability of A given that B has occurred. It restricts attention to the outcomes within event B.

Independent events

Events are independent when P(A∩B)=P(A)·P(B), equivalently P(A|B)=P(A). Knowing one occurred does not change the other's probability.

Addition rule

P(A∪B)=P(A)+P(B)−P(A∩B). For mutually exclusive events the intersection probability is zero, so the probabilities simply add.

Relative risk

The ratio of the risk of an outcome in an exposed group to the risk in an unexposed group. A value of 1 means no difference in risk.

Margin of error

For a proportion, m=z·√(p̂(1−p̂)/n). The margin is largest when p̂=0.5 and shrinks with the square root of sample size.