d
Leave the window with the summary information open and continue with Exercise 10.12.

Exercises
503
10.12
Applet Exercise
Refer to Exercise 10.11. Change
α
to .1 but keep
H
0
:
p
=
.
5
,
H
a
:
p
?=
.
5 and
the true value of
p
=
.
6. Simulate at least 200 tests when
n
=
15. Repeat for
n
=
30
,
50, and
100. Click on the button “Show Summary.” You will now have two summary tables (it might
be necessary to drag the last table from on top of the first). Compare the error rates when tests
are simulated using 15, 30, 50, and 100 trials.
a
Which of the two tests
α
=
.
05 or
α
=
.
10 gives the smaller simulated values for
β
, using
samples of size 15?
b
Which gives the smaller simulated values for
β
for each of the other sample sizes?
10.13
Applet Exercise
If you were to repeat the instructions of Exercise 10.10, using
n
=
100 instead
of
n
=
30, what would you expect to be similar? What would you expect to be different?
10.14
Applet Exercise
Refer to Exercise 10.9.Setup the appletto test
H
0
:
p
=
.
1versus
H
a
:
p
< .
1
by clicking the radio button “Lower” in the line labeled “Tail” and adjusting the hypothesized
value to .1. Set the true value of
p
=
.
1,
n
=
5, and
α
=
.
20.
a
Click the button “Draw Sample” until you obtain a sample with zero successes. What is
the value of
z
? What is the smallest possible value for
z
? Is it possible that you will get a
sample so that the value of
z
falls in the rejection region? What does this imply about the
probability that the “large sample” test procedure will reject the null hypothesis? Does this
result invalidate the use of large sample tests for a proportion?
b
Will the test from part (a) reject the true null approximately 20% of the time if we use
n
=
10? Try it by simulating at least 100 tests. What proportion of the simulations result
in rejection of the null hypothesis?
c
Look through the values of ˆ
p
in the table under the normal curve and identify the value of
ˆ
p
for which the null is rejected. Use the tables in the appendix to compute the probability
of observing this value when
n
=
10 and
p
=
.
1. Is this value close to .2?
d
Is
n
=
100 large enough so that the simulated proportion of rejects is close to .2? Simulate
at least 100 tests and give your answer based on the simulation.
10.15
Applet Exercise
Refer to Exercise 10.10. Click the button “Clear Summary” to delete the
results of any previous simulations. Change the sample size for each simulation to
n
=
30
and set up the applet to simulate testing
H
0
:
p
=
.
4 versus
H
a
:
p
> .
4 at the .05 level of
significance.
a
Click the button “Clear Summary” to erase the results or any previous simulations. Set
the real value of
p
to .4 and implement at least 200 simulations. What is the percentage
simulated tests that result in rejecting the null hypothesis? Does the test work as you
expected?
b
Leave all settings as they were in part (a) but change the real value of
p
to .5. Simulate
at least 200 tests. Repeat when the real value of
p
is .6 and .7. Click the button “Show
Summary.” What do you observe about the rejection rate as the true value of
p
gets further
from .4 and closer to 1? Does the pattern that you observe match your impression of how
a good test should perform?