7-2: Estimating Population Proportions
Notation:
p = population proportion
p̂ = sample proportion
n = sample size
q=1−p
Central Limit Theorem for Proportions: As the sample size gets
larger, the sampling distribution of p̂q
approaches normal with mean
q
pq
=
µp̂ = p and standard deviation σp̂ = p(1−p)
n
n.
7-2: Estimating Population Proportions
Example: According to Mars Chocolate North America, they produce
plain M&M’s so that 24% are blue. In our sample from class, we
found that 168 out of 811 plain M&M’s were blue. Find the
probability of getting a sample of size 811 with less than 21% of the
M&M’s being blue.
7-2: Point Estimates
A point estimate is a single value used to give an estimate for the
value of a population parameter.
Example: From our sample of plain M&M’s we calculated the point
estimate p̂ = .21 representing the percent of M&M’s that are blue.
Sample statistics such as x̄, x̃, s, p̂, etc. are all point estimates.
7-2: Confidence Intervals
A confidence interval is a range of values that give an estimate for
the value of a population parameter.
It can be written in one of the following forms:
• statistic − E < parameter < statistic + E
• statistic ± E
• (statistic − E, statistic + E)
Where E is the margin of error, that is, the maximum likely
difference between the observed sample value and the population
parameter.
7-2: Confidence Intervals
A confidence interval must be accompanied by the confidence level
which is the probability that the interval actually contains the
population parameter.
The confidence level gives the success rate of the procedure used to
construct the confidence interval.
Associated with each confidence level is a critical value, zα/2 , that is
the z score separating an area of α/2 in the right tail of the standard
normal distribution.
7-2: Estimating Population Proportions with Confidence Intervals
A confidence interval with confidence level 1 − α for the proportion p
of a population based upon a simple random sample of size n with
sample proportion p̂ is given by:
p̂ − E < p < p̂ + E
p̂ ± E
(p̂ − E, p̂ + E)
Where E = zα/2
q
p̂q̂
n.
7-2: Estimating Population Proportions with Confidence Intervals
Example: In our sample from class, we found that 168 out of 811
plain M&M’s were blue. Find the 95% confidence interval for the
proportion of all M&M’s that are blue. (Find the 99% confidence
interval.)
7-2: Finding the Sample Size to Estimate a Population Proportion
If we solve E = zα/2
q
p̂q̂
n
for n we obtain
n=
z2α/2 p̂q̂
E2
• This allows us to determine the sample size needed to estimate a
population proportion with a confidence interval.
• Here we use the best available information to estimate p̂. If we
don’t have an estimate for p̂ we can use p̂ = .5.
• If the computed sample size is not a whole number, round up to the
next larger whole number.
7-2: Finding the Sample Size to Estimate a Population Proportion
How many M&M’s must be sampled in order to be 95% confident
that the sample percentage is in error by no more than one percentage
point?