Sample Mean Distributions Math 4030 – 8b Sample Mean Distributions 9/13/2018

Sample in terms of iid RV: Population: a random variable X with certain distribution (discrete or continuous); Sample (of size n): n independent random variables that have the same (identical) distribution as X. A random sample of size n can be viewed as an n-dimensional random vector of which all components have the independent and identical distribution, called population or underlying distribution. 9/13/2018

Parameter vs. Statistic Parameters are numbers that summarize data for an entire population. Statistics are numbers that summarize data from a sample, a subset of the entire population. Sample statistics are random variables, while population parameters are not. 9/13/2018

An example of sample means: Population: S = {0, 1, 2, 3, 3, 5, 6, 9} Consider random samples of size 3, the following samples are equal likely to be formed: {012, 123, 013, 015, 016, 019, 023, 123, 025, 026, 029, 033, 035, 036, 039, 035, 036, 039, 056, 059, 069; 123, 123, 125, 126, 129, 133, 135, 136, 139, 135, 136, 139, 156, 159, 169; 233, 235, 236, 239, 235, 236, 239, 256, 259, 269; 335, 336, 339, 356, 359, 369; 356, 359, 369; 569} 9/13/2018

Sampling distribution of the Mean X has any distribution with the mean µ and standard deviation . Let be the sample mean from an (independent) sample of size n. Then 60

Finite Population Correction Factor If the population is finite, sample variables cannot be independent. However we still have, Finite Population Correction Factor 60

Law of Large Number: X1,X2,…, Xn is a sample from a population with (finite) mean  and (finite) variance 2, then for any arbitrary (small and) positive number , where is the sample mean. (Long-run) relative frequency and probability. 9/13/2018

The Central Limit Theorem X has any distribution with the mean µ and standard deviation . Let be the sample mean from an (independent) samples of size n. Then if n is large. Further more, if the population variance is unknown, we may use the sample standard deviation. i.e. if n is large. (n ≥ 30) 60

Sample mean distribution If the population is normally distributed with known mean and variance, then the sample mean is normally distributed. For large sample (at least 30), the sample mean is approximately normally distributed (Central Limit Theorem).  unknown and n < 30?

If population is normally distributed with unknown variance, Where S is the sample standard deviation, and t(n-1) is the t-distribution with degree of freedom n-1.

Distribution of Sample Means: Use z-Table Use t-Table for n < 30 and z-Table for n ≥ 30.