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7.2 Estimating a Population Proportion
When we don’t know the population proportion, p, the best estimate is the sample
proportion
p (p hat). But, just how good is it?
A confidence interval is a range of values that is likely to contain the true value of the
population parameter (p in this case).
The degree (or level) of confidence is the probability 1 – α that the confidence interval
contains the true value of the population parameter. It is also the percentage of times that the
confidence interval contains the population parameter.
Common choices for α are 0.01, 0.05, 0.
Notation:
α = the total area under both tails
α/2 = the area under one tail
Z α/
= the critical value which corresponds to the point that
separates an area of α/2 in the right tail
More Notation:
p =
x
n
= the best point estimate
q = 1 –
p
E =
/ 2
pq
Z
n
α
= Margin of Error
The Confidence Interval is:
pˆ
pˆ
Finding z
α/ 2
for a 95% Confidence Level
- z
α/ 2
= –1.
z α/ 2
= 1.
Critical Values
α/ 2 = 2.5% =.
α = 5%
Procedure for constructing a confidence interval for p
- Select a confidence level, and express it as a decimal c. (Use 0.95 unless given)
- Compute α = 1 – c and α/
- Find Z α/
from table A–
- Compute E using E =
/ 2
pq
Z
n
α
- Compute the upper and lower limits
p ± E and write the confidence interval as
p – E < p <
p + E (round to 4 decimal places)
Example:
A random sample of 600 homes produced 318 home owners who admitted to owing
at least one gun.
a. Find a 98% confidence interval for the proportion of home owners
who own a gun.
n = 600
x = 318
p = 318/600 =.
q = 1–.53 =.
CL =.
α = 1 – .98 =.
α/2 =.
Z α/
= 2.
E 2.
.53 – .0475 < p < .53 +.
.4825 < p <.
We are 98% confident that this interval
contains the true proportion.
b. Based on the confidence interval, does it appear than 50% of home owners own
a gun? Why or why not?
Answer: Yes because .50 is contained in the confidence interval.
c. Based on the confidence interval, does it appear that more than 60% of home
owners own a gun? Why or why not?
Answer: No because 0.60 is greater than the confidence interval values. (Note that if
you say no because 0.60 is not contained in the confidence interval you will not
receive full credit.)
.
