Sample Size Determination

CI General Form, where E is the margin of
error is:

Solving for n gives:
Example: Estimating Mean Waiting Time
in the Emergency Room.

Waiting time in the ER during weekends is
Normal(μ=35 minutes, σ=9.5 minutes).

How can I estimate the true waiting time
within 1 minute with the 95% Confidence?
The Sampling Distribution
of the Sample Proportion


Recall:

Normal Curve Approximation to Binomial
distribution

Sampling Distribution of the Sample Mean

Central Limit Theorem
New Result:

Sampling Distribution of the Sample Proportion: if
np>=5 and n(1-p) >=5
Facts: According to a spring 2009 poll of more than
2,200 college students across 40 colleges and
universities:

85 percent of students reported feeling stressed on a daily
basis.

Academic concerns like school work and grades, with 77
percent and 74 percent, maintain their positions as the top
drivers of student stress, even over financial.

Six out of 10 students report having felt so stressed they
couldn’t get their work done on one or more occasions.

Since starting college, over 70 percent of students have
not considered talking to a counselor to help them deal
with stress or other emotional issues.
Example


Assume that the true proportion of students
who feel stressed on a daily basis 85%.
What is the probability that in a random poll
of 100 students more than 50% are feeling
stressed on a daily basis?
Example

According to MA 115 B1 Intro Survey, 52 out of
108 students reported feeling stressed at the
beginning of the semester.

What is the approximate probability to obtain the
sample proportion of 48% or lower, if the true
proportion of all students who feel stressed during
the semester is 85%?
Confidence Intervals for p
The assumptions required for CI for a
population proportion to be valid:
 the sample size n is large enough
np̂

5
n 1 pˆ
5
the data are a random sample from that
population.
Confidence Intervals for p
Type of the Formula
General Formula for Confidence
Interval for the Population
Proportion p
Formula for Approximate
Confidence Interval for the
Population Proportion p
Formula for Conservative
Confidence Interval for the
Population Proportion p
Note: here p=0.5 is used to
compute the standard error,
Formula
pˆ Z 1
pˆ Z1 (
p(1 p)
n
( / 2)
/ 2)
pˆ Z1 (
pˆ (1 pˆ )
n
1
/ 2)
2 n
Example
The proportion of adults that believe in love
at first sight. Assume for the sample of 100
people 40 will say they do believe.

(a) Compute an approximate 95%
confidence interval.
Example
The proportion of adults that believe in love
at first sight. Assume for the sample of 100
people 40 will say they do believe.

(b) Compute a conservative 95% confidence
interval.
Sample Size Computaion
Formula for the Sample Size
required to produce an estimate
for the population proportion p
Formula for the Approximate
Sample Size. If the population
parameter p is unknown, the
sample estimate can be used
instead
Formula for the Conservative
Sample Size (p=0.5)
n
n
Z1 (
p (1 p )
Z1 (
n 0.25
E
E
2
/ 2)
/ 2)
E
pˆ (1 pˆ )
Z1 (
2
2
/ 2)
Example
The proportion of adults that believe in love
at first sight. Assume for the sample of 100
people 40 will say they do believe.

(d) How many people do we need to survey
in order to estimate the true population
proportion with 5% accuracy? (E=0.05)