population
The entire group of individuals in which we are interested but can't usually assess directly
sample
The part of the population we actually examine and which we do have data
parameter
number describing a characteristic of the population.
- usually unknown
statistic
number describing a characteristic of a sample
- often used to estimate an unknown population parameter
Law of large numbers
Draws observations at random from any population with finite mean. As the number of observation drawn increase, the mean of the observed values get closer and closer to the mean of the population.
sampling distribution
The sampling distribution of a statistic is the distribution of all possible values. Taken by the statistic when all possible samples of a fixed size n are taken from the population
- the sampling distribution of a statistic us the probability distributio
simulating sampling distribution
- Take random sample of size n from a population
- calculate the sample average go then observation
- repeat the procedure m times
- make a histogram of the m sample average
- this shows the approximate sampling distribution of the sample average
mean and standard deviation of the sample mean
- suppose a population has parameter given by the mean and standard deviation
- sample mean is the statistic used to estimate u then :
- mean of xbar = x
- standard deviation of xbar = sigma/squareroot of n
Estimation of mean
since mean of xbar = x
- the sample mean is called on unbiased estimator for u
- is repeated sampling and averaging of the sample means obtained will result in the value u.
sampling distribution of a sample mean
if individual observations have to the N(u,o) distribution, then the sample mean xbar of an SRS of the size n has the N(u,o/squareroot of n) distribution
sample mean
- the means of random samples are less variable than individual observations
- means of random samples are more normal than individual observation
large samples
- not always attainable
- the cost, difficulty or preciousness of what is studied limits drastically any possible sample size
- time of operations
Central limit theorem
- draws and single random sample of size n from any population with the mean u and finite standard deviation o when is large, the sampling distribution of the sample mean is approximately normal
- allows us to use normal probability calculations to answer
sample size
- how large? depends on the population distribution more observation are required if the population distribution is far from normal
- s sample size of 2s is generally enough to obtain s normal sampling distribution from a strong skewness or even mild outl