Utilize formulas to compute probabilities for the Poisson distribution, Po(\(\lambda\)).
Recognize that if X ~ Po(\(\lambda\)), then both the mean and variance of X are equal to \(\lambda\).
Grasp the significance of the Poisson distribution in modeling random events, and apply the Poisson distribution as a suitable model.
Apply the Poisson distribution as an approximation to the binomial distribution when appropriate [knowing the conditions where n is large and p is small, typically when n > 50 and np < 5].
Utilize the normal distribution, with continuity correction, as an approximation to the Poisson distribution when appropriate [knowing the condition that \(\lambda\) is large, typically when \(\lambda\) > 15].
Linear Combinations of Random Variables
Use, when solving problems, the results that:
E(aX + b) = aE(X) + b and Var(aX + b) = a2 Var(X)
E(aX + bY) = aE(X) + bE(Y)
Var(aX + bY) = a2 Var(X) + b2 Var(Y) for independent X and Y
If X has a normal distribution then so does aX + b
If X and Y have independent normal distributions then aX + bY has a normal distribution
If X and Y have independent Poisson distributions then X + Y has a Poisson distribution
Continuous Random Variables
Comprehend the concept of a continuous random variable, and remember and apply properties of a probability density function [for density functions defined over a single interval, recognizing that the domain may be infinite].
Apply a probability density function to solve problems related to probabilities and to compute the mean and variance of a distribution. This includes determining the location of the median or other percentiles of a distribution by directly considering the area using the density function. Explicit knowledge of the cumulative distribution function is not required.
Sampling and Estimation
Recognize the difference between a sample and a population, and understand the importance of randomness in sample selection.
Describe in simple terms why a specific sampling method may be inadequate. This includes a basic understanding of the use of random numbers in generating random samples. Knowledge of specific sampling methods, such as quota or stratified sampling, is not necessary.
Understand that a sample mean can be considered as a random variable, and apply the facts that the expected value of the sample mean E(\(\overline{X}\)) = \(\mu\) and the variance of the sample mean Var(\(\overline{X}\)) = \(\frac{\sigma^2}{n}\).
Use the fact that \(\overline{X}\) has a normal distribution if X has a normal distribution.
Apply the Central Limit Theorem as needed [only an informal understanding of the Central Limit Theorem (CLT) is necessary; for large sample sizes, the distribution of a sample mean is approximately normal].
Compute unbiased estimates of the population mean and variance from a sample, utilizing either raw or summarized data. A basic understanding of the term "unbiased" is sufficient, such as recognizing that although individual estimates may vary, the process provides an accurate result on average.
Calculate and interpret a confidence interval for a population mean in situations where the population is normally distributed with a known variance or when a large sample is employed.
Calculate an approximate confidence interval for a population proportion from a large sample.
Hypothesis Tests
Comprehend the concept of a hypothesis test, distinguish between one-tailed and two-tailed tests, and understand the terms null hypothesis, alternative hypothesis, significance level, rejection region (or critical region), acceptance region, and test statistic. Interpret the outcomes of hypothesis tests in the context of the questions being addressed.
Develop hypotheses and conduct a hypothesis test for a single observation from a population with a binomial or Poisson distribution, employing:
Direct calculation of probabilities
A normal approximation to the binomial or Poisson distribution, when suitable
Develop hypotheses and conduct a hypothesis test regarding the population mean in situations where the population is normally distributed with a known variance or when a large sample is utilized.
Comprehend the concepts of Type I error and Type II error in the context of hypothesis tests.
Compute the probabilities of committing Type I and Type II errors in specific situations involving tests based on a normal distribution or direct evaluation of binomial or Poisson probabilities.
