4.2 Statistics and probability - AHL content

• Design valid surveys/questionnaires and choose relevant variables/data.

• Use unbiased, structured and precise questioning.

• Categorize numerical data for chi-square tests with suitable degrees of freedom.

• Distinguish reliability and validity; know test-retest, parallel forms, content and criterion-related validity.

• Use technology for least-squares regression curves: linear, quadratic, cubic, exponential, power and sine.

• Use residual sum of squares and coefficient of determination R^2 to evaluate fit.

• Know R^2 alone is not enough to choose a model.

• Use linear transformations of random variables: E(aX+b)=aE(X)+b and Var(aX+b)=a^2Var(X).

• Use expected value and variance for linear combinations of independent random variables.

• Use sample mean and unbiased sample variance as estimators.

• Linear combinations of independent normal random variables are normal.

• Use distribution of sample mean: Xbar ~ N(mu, sigma^2/n) for normal populations.

• Use central limit theorem; n>30 is sufficient in examinations.

• Construct confidence intervals for the mean of a normal population.

• Use normal distribution when sigma is known and t-distribution when sigma is unknown.

• Interpret confidence intervals in context.

• Use Poisson distribution, mean and variance.

• Sum of independent Poisson variables is Poisson.

• Choose between normal, binomial and Poisson models based on context.

• Use critical values and critical regions.

• Test population means with normal or t distributions, including paired/unpaired samples.

• Test proportions with binomial distribution and population means with Poisson distribution.

• Test whether population correlation rho is zero; know Type I and II errors and their probabilities.

• Use transition matrices, powers and state matrices: s_n=T^n s_0.

• Use transition diagrams for discrete dynamical systems.

• Work with regular Markov chains, steady state and long-term probabilities.

• Recognize steady state as eigenvector for eigenvalue 1.