Hypothesis Test for a Population Proportion,
Ho:
p
p  p0

H a:
1.
2.
3.
p  p0
p  p0
p  p0

T.S.
pˆ  p0
z
 pˆ
R.R. For a probability of a Type-I error


1. Reject Ho if z > z

2. Reject Ho if z < -z
3. Reject Ho if z < -z/2 or if z > z/2

Note: Under H0,
 pˆ 
p0 (1  p0 )
n
Assumptions:
1. A random sample is selected from a population
2. The sample size is sufficiently large ( np0 (1 
proportion
p̂
p0 )  10 ) such that the sampling distribution of the sample
is approximately normal.
Example State DMV records indicate that of all vehicles undergoing emissions testing during the previous year, 70%
passed on the first try. A random sample of 200 cars tested in a particular county during the current year yields 160 that
passed on the initial test. Does this suggest that the population proportion for this county during the current year differs
from the previous statewide proportion? Conduct hypothesis test using   .05.
Test and CI for One Proportion
Test of p = 0.7 vs p not = 0.7
Sample
1
X
160
N
200
p̂
0.800000
95% CI
(0.744564, 0.855436)
Z-Value
3.09
P-Value
0.002
Example A university library ordinarily has a complete shelf inventory done once
every year. Because of new shelving rules instituted the previous year, the head
librarian believes it may be possible to save money by postponing the inventory.
The librarian decides to select at random 1000 books from the library’s collection
and have them searched in a preliminary manner. If evidence indicates strongly
that the proportion of misshelved or unlocatable books in the population is less than
.02, then the inventory will be postponed.
Among the 1000 books searched, 15 were misshelved or unlocatable. Conduct
hypothesis test and advise the librarian what to do (use   .05. ).