175
Math 116 – 02: Final Examination
Spring 2012
Name:
1. Determine whether each of the following statements is true or false.
(3 points each)
(a) Disjoint events are always independent.
(b) In a hypothesis test, if we reject H0, there is a chance for a type I error.
(c) In a hypotheses test, the p-value is the probability that H0 is true.
(
) () ()
(d) For mutually exclusive events, P A and B = P A × P B .
(e) The test statistic for a hypothesis test is a numerical value summarizing the sample that
has been collected.
2. Suppose we wanted to construct a 95% confidence interval for the average temperature of
people who contract a new strain of influenza with a margin or error of 0.10. Suppose also that
other studies have supported that the standard deviation of human body temperature is 0.45.
How many subjects should be chosen at random for study? (8 points)
3. Suppose we wanted to construct a 95% confidence interval for the proportion of people in a
certain city that are pleased with how their local government is representing them with a
margin of error of no more than 2 percentage points. How many people should be chosen from
this community at random for the study? (8 points)
4. A certain company sells three models of digital cameras (X, Y, and Z). By studying their sales
over a long period of time, it was found that of all cameras sold 20% are model X, 30% are
model Y, the rest are model Z. When a camera is purchased, an extended warranty is available.
Of those that bought model X, 40% purchased the extended warranty. Of those that bought
model Y, 45% purchased the extended warranty. Of those that bought model Z, 60% bought an
extended warranty. Define event W as “person bought an extended warranty”. Find the
following probabilities and explain what each means for a randomly selected customer at this
company. (5 points each) HINT: A tree diagram may be helpful.
( )
(a) P W
(
(b) P Y W C
)
( (
(c) P W X or Z
((
)
))
(d) P X or Y W C
)
5. An investigation of several large universities revealed that 38% of dorms have refrigerators (R),
60% have TV’s (T), and 21% have both. One dorm room is randomly selected. Find the
following probabilities and explain what each means in context. (5 points each)
HINT: A Venn diagram may be helpful.
(
(a) P R and T C
(
)
(b) P RC and T C
)
Cö
æ
(c) P ç R or T and R and T ÷
è
ø
(
)
( )
(d) P R T
(
(e) P T RC
)
(
)
6. A company has developed a new battery with life span of 4200 hours with a standard deviation
of 72 hours. What is the probability that in a sample of 100 such bulbs, the average life span is
less than 4175 hours? (8 points)
7. A company that manufactures computer chips believes that only 2% of their chips are defective.
Suppose that this is true.
(a) What is the smallest sized sample for which we could say that the distribution of the
variable p̂ is normally distributed? (4 points)
(b) What is the probability that in a sample of 1000 of this company’s computer chips, more
than 3% are defective? (6 points)
8. A certain state’s Department of Education randomly selected 500 students and found that 58 of
them attended private school. Construct a 98.5% confidence interval for the proportion of all
students in this state that attend private school. (5 points)
9. The math SAT scores of 5 randomly selected incoming freshman at a certain university are 540,
470, 590, 610, and 650. Construct a 96% confidence interval for the average math SAT scores
of all incoming freshman at this university. Do not forget to verify that the process is valid.
(8 points)
10. For each of the following hypothesis tests, calculate the p-value and give the appropriate
conclusion at the  = 0.05 level. (6 points each)
(a)
H0 : m = 850
Ha : m ¹ 850
t = 1.875
n = 10
(b)
H0 : p1 = p2
Ha : p1 < p2
z = -1.95
(c)
H0 : p1 = p2 = p3 = p4 = p5 = p6 = 0.16
Ha : not H0
c 2 = 20.52
11. Many math professors believe that their students are studying, on average, less than the
recommended 12 hours per week for their math courses. A representative sample of 40
students was selected. The average amount of time spent studying by this group was 10.4
hours with a standard deviation of 2.45 hours. Use a 95% confidence interval to test whether
the professors’ beliefs are accurate. (10 points)
H0:
Ha:
95% CI:
Conclusion:
Validity:
12. A certain car manufacturer is planning to offer its newest model car in three possible colors:
silver, blue, and green. The manager of the company believes that 50% of the cars sold will be
silver, 30% will be blue and 20% will be green, and so production is planned accordingly. An
independent researcher is curious if these proportions are correct, so 200 people planning to
buy a new car in the next year are randomly chosen and asked which color they would choose.
The results are shown below. Check if the manager’s proportions are accurate by testing the
relevant hypotheses at the  = 0.05 level. (10 points)
Color
Silver
Blue
Green
# People
88
80
32
H0:
Validity:
Ha:
Test Statistic:
p-Value:
Conclusion
13. The numbers below give summary data for the average ACT scores for incoming freshman at a
certain university for two groups: those receiving football scholarships and those not receiving
football scholarships. (5 points each)
x
s
n
Scholarship
21.86
2.84
44
No
Scholarship
24.75
3.29
31
(a) Construct a 98% confidence interval for the difference in average ACT scores for
football scholarship students and non-football scholarship students.
(b) Based on your answer to (a), would you conclude that average ACT scores are the same
for these two populations or not. Support your answer.
14. A legislator is curious whether a majority of the voters in her district favor a law that would
reduce the legal blood alcohol level that defines “legally drunk”. She has her staff collect data
for analysis. They find that of 260 randomly selected voters in the district, 142 would favor
such a law. Does this data provide sufficient evidence that a majority (more than 50%) of the
voters are in favor of such a law? Test the relevant hypotheses at the  = 0.05 level. (10 points)
H0:
Validity:
Ha:
Test Statistic:
p-Value:
Conclusion:
15. The question of whether or not taking aspirin reduces the chance of having a heart attack has
been of interest for some time. In one particular experiment, a very large sample was randomly
selected from a pool of people of comparable age and health. One group was given aspirin
every day and the other a placebo. Of the 11034 people that were given a placebo daily, 189
had a heart attack at some point while only 104 of the 11037 people given aspirin had a heart
attack. Is there significant evidence to support that the proportion of people that have a heart
attack is lower for those taking aspirin than for those that do not? Test the relevant hypotheses
at the  = 0.05 level. (10 points)
H0:
Ha:
Test Stat:
P-Value =
Conclusion:
Validity: