Suppose we want to test the hypothesis
0
:
1
0
H
, vs
1
:
0
a
H
If we want
01
.
0
We would reject the null when:
1.
99
.
0
1
1
)
(
t
b
SE
b
2.
1
0.995
1
(
)
b
t
SE b
3.
99
.
0
1
1
)
(
t
b
SE
b
4.
99
.
0
1
1
)
(
z
b
SE
b
2.9
Suppose we want to test the hypothesis
0
:
1
0
H
, vs
1
:
0
a
H
If we want
01
.
0
We would FAIL TO reject the null when:
1.
1
0.99
1
(
)
b
t
SE b
2.
1
0.01
1
(
)
b
t
SE b
3.
1
0.99
1
(
)
b
t
SE b
4.
1
0.95
1
(
)
b
t
SE b

Practice Final
Stat 311 Spring 2015
5
Problem 4.
Estimating proportions
The Department of Statistics estimates that 30% of students at the University of Washington will have taken STAT 311 by
the time they graduate.
You believe it is less than 30%, and decide to take a survey of graduating seniors to estimate
the true proportion.
Your friend gets you a list of the graduating seniors and their email addresses, and you randomly
select a sample of n=100, email them, and ask them to report on a Catalyst survey whether they have taken STAT 311:
25 of the 100 seniors report that they have.
Set up a 90% CI for the estimated proportion of seniors who have taken STAT 311.
4.1
Symbolic representation:
4.2
With plug-in values:
State the null and the general alternative hypotheses. What is the approximate distribution of the sample
proportion under the null hypothesis?
(Specify the form of the null distribution, the mean and the standard
deviation, you do not need to solve for numerical value of the standard deviation).
4.3
𝐻
0
𝑣?. 𝐻
𝑎
4.4
Distribution of
𝑝̂
under
𝐻
0
Regardless of your answers above, set up a hypothesis test of
H
0
:
p
= 0.30 vs.
H
a
:
p
< 0.3
using α=0.05
4.5
Test statistic in symbols
4.6
Test statistic with values
plugged in
4.7
Is this a
one-tail
or
two-tail test?
(circle one)
:

Practice Final
Stat 311 Spring 2015
6
4.8
We would reject the null when our test stat is
greater than/
less than/
equal to
_______.
(circle one)
(fill in value)
You want to calculate the power of this test against your alternative hypothesis,
that the true proportion is 20%.
To do
this, you will need to calculate pcrit
, the critical val
ue of p under the null hypothesis for α=0.05.
Set up the formula you would use to solve for pcrit .
You do not need to solve for the answer.
4.9
Formula for pcrit
4.10
With plug in values
4.11
Say the value of pcrit turns out to be 0.22.
Draw the distributions of the null and the alternative hypothesis,
and label each.
Label the values of
p0 , pa , and pcrit on the x axis (IMPORTANT
–
EACH OF THESE LABELS IS
WORTH 1 POINT).
Shade in the appropriate area that represents the power of the test. The drawing does not have
to be the right scale.
Set up the formula for calculating power, and plug in the values.
4.12
Formula for the power of this test:
4.13
With plug in values:

Practice Final
Stat 311 Spring 2015
7