Download Distribution of Sample Proportions

QMS 202 Distributions of Sample Proportions
Consider the Population:
Y,N,N,N,N,N,N,N,N,N ...
10% Y’s and 90% N’s
Define p as the proportion of Y’s
QMS 202 Distributions of Sample Proportions
Now consider all possible samples of size n=100 which can be taken from the population above where
the 1st item selected is replaced before the 2nd item is selected etc. [This sampling procedure is
equivalent to sampling from an infinite population].
The Count the number of Y’s in the samples this forms a Binomial Distribution.
Distribution of Sample Counts
X
(# Y’s)
......
6
7
8
9
10
11
12
13
14
.....
100
P(X)
.06
.09
.11
.13
.13
.12
.10
.07
.05
Which is approximately normal with the same mean and
standard deviation as the binomial distribution.
Using Binomial formula with p = .1
and n=100
Note: All binomial distributions where
approximately normal.
are
QMS 202 Distributions of Sample Proportions
Now consider the Distribution of Sample Proportions, p, created by dividing the counts, X, by n (100).
p Distribution
Sample
Probability( p)
Proportions
0
......
6/100=.06
.06
.07
.09
.08
.11
.09
.13
.10
.13
.11
.12
.12
.10
.13
.07
.14
.05
.....
1
Which is approximately normal with a mean of
provided
and
and standard deviation
are both $5 are approximately normal.
QMS 202 Distributions of Sample Proportions
35% of students have cell phones. If a sample of 50 students is randomly selected what is the probability
that the proportion of students in the sample that have cell phones will be less that 30%? (Note that this
is equivalent to the binomial question : What is the probability that less than 15 of the students sampled
have cell phones?)
p - proportion of students in the sample that have a cell phone.
Since
, p’s are normally distributed.
Using the standard normal distribution we get