Workshop Statistics

1
Workshop Statistics

• Activity 16-3 continued

2
Since these sample proportions follow
approximately a normal distribution, the
empirical rule establishes that about 95 of the
sample proportions fall within two standard
deviations of the mean of these sample
proportions.
3
Remembering that you found the mean and standard
deviation of these sample proportions for a
sample size of 75, double the standard deviation.
Then subtract this value from the mean and also
add this value to the mean. Record the results.
4
Use your calculator to count how many of the
sample proportions fall within this interval.
What percentage of the 500 sample proportions is
this? Is this percentage close to the 95
predicted by the empirical rule?
5
If each of the 500 imaginary students would
subtract this value (twice the standard
deviation) from his or her sample proportion and
also add this value to her or his sample
proportion, about what percentage of the
students intervals would contain the actual
population proportions of .45?
6
This activity reveals that if one wants to be
about 95 confident of capturing the population
proportion within a certain distance of ones
sample proportion, that distance should be
about twice the standard deviation of the
sampling distribution of sample proportions.
7
One need not use simulations to determine how
sample proportions vary from sample to sample.
An important theoretical result affirms what your
simulations have suggested about the shape,
center, and spread of this distribution
8

• Central Limit Theorem (CLT) for a Sample
  Proportion
• Suppose that an SRS of size n is taken from a
  large population in which the true proportion
  possessing the attribute of interest is ?. Then
  the sampling distribution of the sample
  proportion is approximately normal with mean
  ? and standard deviation

9
This approximation becomes more and more accurate
as the sample size n increases, and it is
generally considered to be valid, provided that
Notice that this result specifies 3 things about
the distribution of sample proportions shape,
center, and spread.
10
If we continue to assume that the populations
proportion of orange candies is ? .45, what
does the theoretical result say about the mean
and standard deviation of the sampling
distribution of sample proportions when the
sample consists of n 25? Do these values come
close to your simulated results (c)
above? Theoretical mean of
values Theoretical standard deviation of
values
11
Repeat the previous question for a sample size of
n75. Compare your theoretical answers to your
simulated results. Theoretical mean of
values Theoretical standard deviation of
values