Chapter 7 Statistics: Sampling Distributions
recall that mu = the population mean, while x bar...
is equal to the sample mean!
SHAPE:
but the shape is what emphasizes the importance of the size of "n". In order to determine the shape of the sampling distribution, we look at size n.
30 = n is just a rule of thumb! not a perfect number
even if X comes from a skewed population, you still can have a pretty normal population distribution!! (it's crazy cool :) ) You just must have a SUFFICIENTLY LARGE sample. Most elementary textbooks suggest the minimum is 30.
why use sampling distributions?
because we are often interested in probabilities about a GROUP instead of an individual random variable. Thus we need to know the distribution of the STATISTIC! For now, the statistic of interest is the SAMPLE MEAN.
Central Limit Theorem
suppose a random sample of size "n" is selected from ANY population! When "n" is sufficiently large, the sampling distribution of x bar has an approximate normal distribution. As "n" gets larger, the approximation becomes better! :) THE POPULATION DOES NOT HAVE TO BE NORMAL!!! :)
we use the STATISTIC...
...to draw conclusions about the PARAMETER!
No-Name Theorem
if a random sample of "n" observations is selected from a NORMAL population, the sampling distribution of x bar will ALSO have a normal distribution! If X is normal, X bar is normal NO MATTER WHAT!
Sampling Distribution
the sampling distribution is the distribution of all possible values that can be assumed by some statistic, computed from samples of the same size randomly drawn from the same population. IT IS THE PROBABILITY DISTRIBUTION OF THE SAMPLE STATISTIC! It describes ALL POSSIBLE VALUES that can be assumed by the statistic!
Sample statistics are RANDOM VARIABLES...
they will vary from sample to sample and thus can be described by a probability DISTRIBUTION. Obv., Parameters are CONSTANTS and thus CANNOT be described by any sort of distribution because they have a FIXED PROBABILITY!
The sampling distribution of x bar aka the Expected Value of x bar
we have interest in a group, so we have a sampling distribution of x bar (the mean of our samples) and we use these samples to draw conclusions about mu (the population mean)! mu = the population mean xbar = the sample mean NOTE: n does not matter!!! ALL SAMPLING DISTRIBUTIONS have mu of bar = mu and the standard deviation of x bar equal to n divided by root n.
we are interested in three characteristics about a given sampling distribution
1. mean 2. variance (or standard deviation) 3. shape (functional form)
as we increase the size of our samples, the sampling distribution becomes more and more normal, EVEN if the population distribution of the random variable...
...is NOT NORMAL! Our rule of thumb is that "n" should be greater than or equal to 30!
steps in constructing a sampling distribution
1. Randomly draw all possible samples of size "n" from a FINITE population of size "N." (if the population is infinite, then do "repeated sampling") 2. Compute the Statistic for each sample 3. List in one column the different observed values of the statistic, and in another column the corresponding frequency of occurrence OR construct a histogram - the PDF will be the curve that is attained by smoothing out the histogram
Properties of the sampling distribution of x bar
1. the mean of the sampling distribution of x bar: mu of x bar = mu (the mean of the sampling distribution of x bar is equal to the original population mean). Remember that the population mean aka "expected value" 2. the standard deviation of the sampling distribution of x bar = standard deviation of the population divided by the square root of "n" 3. what about the shape?? normal if "n" is "sufficiently large"
statistic
a numerical value based on the SAMPLE
Parameter
a numerical value based on the population
