Download Sampling Distribution - New Paltz Central School District

Sampling Distribution
The distribution you would get from gathering a statistic on all possible samples of a
certain size from a population. It is understood that each sample may produce a different
statistic but the sampling distribution shows how all the samples behave.
The two types of sampling distributions we are going to study are:
1) THE SAMPLING DISTRIBUTION FOR PROPORTIONS
2) THE SAMPLING DISTRIBUTION FOR MEANS
Based on the fundamental theorem of statistics (sometimes known as the central limit
theorem), it was proven that the above mentioned sampling distributions are
approximately normally distributed (if certain conditions are true). That’s a BIG
IDEA . . . .the population may or may not be distributed normally, or we may not even
know about the population’s distribution, but the sampling distribution is normal
We know from past experience that to use a normal model to describe a distribution we
need to know the mean and standard deviation:
THE SAMPLING DISTRIBUTION FOR PROPORTIONS:
To approximate this with a normal distribution we need to first check certain
conditions, which are
a. is the sample a random sample from the population (this helps ensure
the responses in the sample are independent)
b. is the sample less than 10% of the population (this helps ensure the
probability of success is constant)
c. are there at least 10 successes and 10 failures expected in the sample
The mean of the sampling distribution (the center) is the mean (center) of the population
The standard deviation of the sampling distribution is
pq
n
THE SAMPLING DISTRIBUTION FOR MEANS
To approximate this with a normal distribution we need to first check certain conditions
(they are the same as the sampling distribution for proportions) except for # of success
and failure
The mean of the sampling distribution (the center) is the mean (center) of the population
The standard deviation of the sampling distribution is
" s tan dar deviation of population"
n
Practice problems: these are all for proportions
1) Suppose a fair coin is tossed 40 times and the proportion of tails is recorded.
a. Is that a sampling distribution?
b. What would the mean and the standard deviation for the sampling
distribution be?
c. Could a normal model describe the sampling distribution?
d. What are the chances of seeing at least 62.5% tails in the 40 tosses
2) Public health statistics indicate that 26.4% of American adults smoke cigarettes.
a. If a random sample of 50 adults were taken, could a normal model help
describe the number of smokers we would expect to get?
b. What is the probability that at least 14 of those randomly selected adults
smoke?
c. If 23 people in that random sample claimed to have smoked would you
re-consider the Public Health Estimate of 26.4%? explain
3) Just before a vote on a school budget, a local newspaper polls 400 voters in an
attempt to predict whether the budget will pass or not. Suppose the budget
actually has a 52% support rate from the voters. What is the probability that the
newspaper’s sample will lead them to predict defeat on the budget?