STAT 515 ā Fall 2001 - Final Exam - Solutions
Part I: Answer three of the following four questions. If you answer more than three I will grade only the first three.
Five points each.
1) Define what is meant by the p-value (or observed significance level) of a test. The p-value is the probability of
observing a statistic as extreme as the one observed, or more extreme, if the null hypothesis is true. -or- The p-
value is the smallest Ī±
Ī±Ī±
Ī±-level at which the null hypothesis would be rejected.
2) The histogram at the right is skewed left We would expect that its mean would be smaller than its median.
3) A fair coin (probability of a head on one flip = 0.5) is flipped 10 times. What is the probability of observing exactly
5 heads? 2461.0)000976562.0(252
10
5.0
!5!5
!10
5
)5.01(
5
5.0
5
10 ā==ā
ļ£·
ļ£ø
ļ£¶
ļ£¬
ļ£
ļ£«
4) A fair coin (probability of a head on one flip = 0.5) is flipped 1,000 times. Using the normal approximation to the
binomial, approximately what is the probability of observing at least 512 heads?
73.0
811.15
5.11
)5.01)(5.0(1000
)5.0(1000)5.0512(
)1(
)( ā=
ā
āā
=
ā
ā
=pnp
npX
Z
We need to subtract the 0.5 for the continuity correction. The answer we want is P(Z>0.73) = 0.5 -
0.2673=0.2327
Part II: Answer every part of the next three problems. Read each problem carefully, and show your work for full
credit. Twenty points each.
1A) State the appropriate null and alternate hypothesis to test if a high score on the exam seems to indicate mild to
moderate traumatic brain injury. Identify any parameters you use in stating the hypotheses.
H0: Āµ
ĀµĀµ
Āµ=40 vs. HA: Āµ
ĀµĀµ
Āµ>40 where Āµ
ĀµĀµ
Āµ is the average test score of all patients with mild to moderate traumatic brain
injuries
B) Test the hypothesis in part a. Report your conclusion at Ī±=0.05 (e.g. Do we accept or reject H0? Is a high test
score associated with traumatic brain injury, or not?)
compare this to 1.717 and reject H0.
The injured do seem to score higher.
C) Construct a 95% confidence interval for the mean test score of patients who suffered mild to moderate traumatic
brain injury.
)41.57,45.39(98.843.48
23
76.20
074.243.48
2=Ā±=Ā±=Ā± n
s
tx
Ī±
D) What assumptions need to be satisfied in order to trust the results you obtained in parts B and C?
The patients need to be a random sample and their test scores need to follow a normal distribution.
95.1
32.4
43.8
23
76.20
4043.48
221 ==
ā
=
ā
=
=ā=
n
s
x
ndf
t
Āµ
