Population and sample means and the central limit theorem

Question 1

probability and statistics

Answer 1

• calculate sample mean or x bar and use it to estimate the position of mean mew
• to know accuracy of this you need to know distribution of X bar (random variable, depends on sample taken)

Question 2

sample mean x bar and population mean mew

Answer 2

these mean should the same but may differ

X bar is an unbiased estimate of mewx - the mean of the sample distribution is the population mean

Question 3

mean and variance of sample mean x bar

implications?

Answer 3

if X bar is the mean of a simple random sample of size n from a population with mean mew and standard deviation sigma, then sample mean = population mean and sigma of sample = sigma/squareroot(n)

implications: X bar is unbiased
SD of sample mean is ~ 1 square root (n) so on average if you have to double the accuracy of your sample mean as an estimate of the unknown population mean, you need to take 4 times as big a sample

Question 4

central limit theorum

Answer 4

variability of sampling distribution of a sample mean decreases as sample size grows

X bar ~ N (mew, sigma/square root (n)) approx. It is exact if the original population is normal

Usually a good approximation for n more than 20

Question 5

If X has mean and SD of mew and sigma, then

Answer 5

1. nX has mean and SD, nmew and n sigma

2. there are some other equations