S3

S3

1. Combinations of random variables

   • If X1 and X2 are independent random variables such that X1 ~ N(u1, o1^2) and X2~N(u2,o2^2) then:
     • X1+X2~(u1+u2, o1^2+o2^2)
     • X1-X2~(u1-u2, o1^2+o2^2)
   • If X1 and X2 are independent random variables such that X1 ~ N(u1, o1^2) and X2~N(u2,o2^2) then:
     • aX1+bX2~(au1+bu2,(a^2)*( o1^2)+(b^2)*(o2^2))
     • aX10-bX2~(au1-bu2,(a^2)*( o1^2)+(b^2)*(o2^2))

2. Sampling

   • Statistically a population is the whole set of items that are of interest.
     • A census observes or measures every member of the population.
       • Advantages: It should give a completely accurate result.
       • Disadvantages: It is very time consuming and expensive. It cannot be used when the testing process is to destruction. The information is difficult to process because there is so much of it.
   • A sample is a selection of observations taken from a sub-set of the population, which is used to fin out information about the population as a whole. this is known a sample survey.
     • Advantages: A sample costs less than a census. Results are obtained quicker for a sample survey rather than for a census. Fewer people have to respond in the sample. There is less data to deal with than in a census.
     • Disadvantages: The data may not be as accurate. The sample may not be large enough to give information about small sub-groups of the population.
   • A random sample is one in which every possible sample size n has an equal chance of being selected
     • In random sampling, each element is given a number to identify it and the numbers of the required elements are selected by using random number tables or other random number generators.
   • A sampling frame is a list identifying every single sampling unit that could be inclined in the sample.
   • In systematic sampling the required elements are chosen at regular intervals from an ordered list.
   • In stratified sampling the population is divided into mutually exclusive strata and a simple random sample is taken from each. The proportion of the strata in the sample is the same as the proportion of the strata in the population.
   • In quota sampling the population is divided into groups ion terms of gender, social class, etc. The number of people in each group is set to try and reflect the group's proportion in the whole population. The interviewer selects the actual sampling units.

3. Estimation, confidence intervals and tests

   • Unbiased estimators
     • X bar = sum of X over n, is an unbiased estimator of the population mean u.
     • S^2 = (sum of X^2 - n(X bar)^2)/(n-1), is an unbiased estimator of the population variance a^2.
   • The Central Limit Theorem
     • sates that if sample size n is large the X= ~ N(u, a^2/n) no matter what the distributin of the population.
   • Confidence intervals
     • A 95% confidence interval for the population mean u is given by:
       • X +- 1.96 * a*(n)^(-1/2)
         • where the 1.96 comes from the normal distribution tables and relates to the confidence level.
   • Hypothesis tests (single sample)
     • The test statistic for a hypothesis test of the population mean u based on a single sample is:
       • z = (xbar - mean)/(a*(n)^(-1/2)
   • Hypothesis tests (two independent samples)
     • The test statistic for a hypothesis test about values of population means ux and uy is:
       • z = ((xbar - ybar) - (ux-uy)) over the square root of (ax^2)/nx + (ab^2)/ny
   • Large samples
     • If the sample size is large then s^2 is often assumed to be equal to a^2.