Mean The mean of random variable x is equal to the expected value E(x). This is the population mean and denoted by μ. A sample mean x may be used to estimate μ.
Varience A measure of the dispersion of a random variable or of a sample. Is equal to E(x2)
Statistic A function of a sample, a quantity calculated from a set of observations. E.g. sample mean and sample varience
Poisson Distribution The discrete probability distribution with probability function given by P(X=r) = (e-λλr)/r! for r=0,1,2,…n where λ is a positive parameter. λ=mean and varience. X~Po(λ)
Continuous Random Variable A quality that takes different numerical values according to the particular experiment. It is continuous if the set of values forms an interval, finite or infinite.
Probability Density Function For continuous random variable x, the p.d.f is:
P(a ≤ x ≥ b) = ∫f(x)dx
Cumulative Distribution Function For random variable x, the c.d.f is:
F(x) = P(X ≤ x)
F(x) = ∑p(xi)
Mode Most frequently occurring observation in a sample or, for grouped data, the group with the highest frequency.
Median Middle observation. ½(n+1)
For p.d.f ∫ f(x) dx = 0.5
Sample Selection of individual members from a population. A collection of sampling units drawn from a sampling frame. It is a random sample it is chosen in the way that every sample of the same size has an equal chance of being selected.
Population Collection of individuals or items
Finite Countable (each member can be given a number)
Infinite Too many to count (impossible to number each)
Countably infinite Lots but still countable (infinite size but can give each a number)
Sampling Units Individual member of a population
Sampling Frame A list of sampling units used in practice to represent a population. Sometimes are the same things
Inference Process of coming to a conclusion about a population based on a sample.
Hypothesis Tests
An assertion about a population. A null hypothesis (usually H0) is a particular assertion that is to be accepted or rejected. The alternative hypothesis (H1) specifies some alternative to H0. To decide whether H0 is to be accepted or rejected , a significance test tests whether a sample taken from the population could have occurred by chance , given the H0 is true.
From the sample, the test statistic is calculated. The test partitions the set of possible values into the acceptance region and the critical (or rejection) region. These depend upon the choice of significance level α , which is the probability that the test statistic lies in the critical region if H0 is true. Often α is chosen to be 5% e.g. if the test statistic falls in the critical region, the null hypothesis H0 is rejected; otherwise the conclusion is that there is no evidence for rejecting H0 and it is said that H0is accepted.
The null hypothesis H0 normally specifies that a certain parameter has a certain value. If the alternative hypothesis H1 says that the parameter is not equal to this value, then the test is said to be two-tailed. If H1 says that the parameter is greater than this value (or less than) the test is one-tailed.
