Chapter 7, continued.... IV. Introduction to Sampling Distributions Suppose you take a second sample of n=30 and calculate your estimators again: s =

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Presentation transcript:

Chapter 7, continued...

IV. Introduction to Sampling Distributions Suppose you take a second sample of n=30 and calculate your estimators again: s = $ To see what would happen if you repeated the sampling process 500 times, see tables 7.5 and 7.6 in the text.

Random variables revisited is a random variable: a numerical description of an experiment. The experiment is the process of selecting a simple random sample. With a large population, you are almost assured that every single time you take a random sample, you’ll get a different value for

Probability distributions revisited Just like any other random variable, has an expected value, a variance, and a probability distribution, which we will begin calling a sampling distribution. Knowledge of this sampling distribution will allow us to make probability statements about how close is to .

V. Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean,.

A. Expected value of E( ) =  This result is proven in the textbook. Thus for the EAI example, the expected value of is $51,800.

B. Standard Deviation of Finite Population: Infinite Population : the standard deviation of the sampling distribution of  : the standard deviation of the population n: the sample size N: the population size This can be used if n/N .05

EAI study n/N= 30/2500 =.012 <.05 so we can assume this is a “large” or infinite population. This is referred to as the standard error of the mean.

An example A survey of library users is taken of 321 people leaving the Indianapolis Central Library. One question asks “How much time did you spend in the library?” The next slide is a histogram that represents the frequency distribution of the variable “SPEND”.

Population parameters:  = minutes,  = minutes. Eric R. Dodge: thanks to John Ottensman at IUPUI. Eric R. Dodge: thanks to John Ottensman at IUPUI.

Sampling distribution for SPEND Now suppose you take a sample of n=8 and calculate. Then do this 50 times and recreate your frequency distribution. One such result is on the next slide.

Mean = minutes, standard error = Eric R. Dodge: thanks to John Ottensman at IUPUI. Eric R. Dodge: thanks to John Ottensman at IUPUI.

Take a larger sample? When we take a sample of n=8 and do it 50 times, we get a mean that is fairly close to the actual mean of SPEND. Do you think the mean of the sampling distribution would be more accurate if we took larger (or more) samples? Let’s try 2500 samples, each with n=8.

Mean of the sampling distribution is 41.25, standard error is Eric R. Dodge: thanks to John Ottensman at IUPUI. Eric R. Dodge: thanks to John Ottensman at IUPUI. Notice how the mean gets closer to the “true” population mean. This is the topic for your next outline.