Stat 312: Lecture 12
Testing on Population Proportion
Moo K. Chung
Feb 20, 2003
Concepts
1. Testing mean µwhen the sample size is large.
H0:µ=µ0vs. H1:µ < µ0
Test statistic: z=¯x−µ0
s/√n.Rejection region for
level αtest: z≤ −zα.
2. Let pbe the proportion of a population with
a specified property. Large-sample level αtest
for
H0:p=p0vs. H1:p < p0
test statistic: z=ˆp−p0
√p0(1−p0)/n .
rejection region: z < −zα.
3. The P-value is the smallest level of significance
at which H0would be rejected.
P-value ≤α→reject H0at level α.
P-value > α →do not reject H0at level α.
The smaller the P-value, the easier to reject
H0.
In-class problems
Example 8.11. 47 out of 102 doctors did not know
the generic name for the drug methadone. Can you
conclude that fewer than half of all doctors know
the generic name for methadone? Test it at 1%
level.
> p<-47/102
> (p-0.5)/sqrt(0.5*0.5/102)
[1] -0.792118
> qnorm(0.01)
[1] -2.326348
Assume the IQ of a dog follows normal. The IQs
of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50,
60, 100, 60. Find the P-value for testing
H0:µ= 100 vs. H1:µ < 100
> t=(mean(x)-100)/(sd(x)/sqrt(10))
> t
[1] -4.205156
> qt(c(0.001,0.002,0.003),9)
[1] -4.296806 -3.834510 -3.572577
> pt(-4.205156,9)
[1] 0.001144577
> t.test(x,mu=100,alternative="less",
conf.level=0.95)
One Sample t-test
data: x t = -4.2036, df = 9,
p-value = 0.001147
alternative hypothesis: true mean is
less than 100
95 percent confidence interval:
-Inf 78.8531
sample estimates: mean of x 62.5
Three options for alternative:less, greater,
two.sided.
Find the P-value for testing
H0:µ= 100 vs. H1:µ6= 100
Self-study problems
Exercise 8.35., Example 8.17., Find the P-value for
testing
H0:µ= 100 vs. H1:µ > 100
Assignment 4.
Due March 13. 11:00am. 8.24., 8.26, 8.28., 8.36.
(a) and (b) only, 8.38.
