Download MATH220 Test 2 Summer 2012 This test has 5 problems which are

MATH220
Test 2
Summer 2012
This test has 5 problems which are worth 100 points. Show your steps in
each problem to receive full or partial credit. Note there is only one correct
answer for multiple choice problems.
Formulas you may need:
x̄−µ
√ .
z = σ/
n
P (A1 andA2 ...andAk ) = P (A1 )P (A2 )...P (Ak ) if A1 , · · · , Ak are all independent.
P (AorB) = P (A) + P (B) − P (AandB).
.
P (A|B) = P (AandB)
P (B)
1. (20 pts) The FAA tells airlines to assume that passengers average 185
pounds in the summer, including clothing and carry-on luggage. But
passengers vary, and a reasonable standard deviation is 30 pounds.
(a) Find the probability that a randomly selected passenger exceeds
200 pounds, i.e., the proportion of passengers who exceeds 200
pounds.
P (x > 200) = P (z > 200−185
) = P (z > 0.5) = 1 − 0.6915 =
30
0.3085.
(b) A commuter plane carries 16 passengers. What is the probability
that the mean weight of the passengers exceeds 200 pounds?
√ ) = P (z > 2.00) = 0.0228.
P (x̄ > 200) = P (z > 200−185
30/ 16
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(c) A Boeing plane carries 144 passengers. What is the probability
that the mean weight of the passengers exceeds 200 pounds?
200−185
√
P (x̄ > 200) = P (z > 30/
) = P (z > 6.00) < 0.0001.
144
2. (25 pts) As suburban gardeners know, deer will eat almost anything
green. In a study of pine seedlings at an environmental center, researchers noted how deer damage varied with how much of the seedling
was covered by thorny undergrowth.
Thorny Cover
Deer Damage
Yes
No
total
None
<1/3
1/3 to 2/3
> 2/3
60
76
44
29
151
158
177
176
211
234
221
205
Total
209
662
871
(a) Estimate P (D), the probability that a randomly selected seedling
was damaged by deer.
209
= 0.2400.
P (D) = 209+662
(b) Find the conditional probabilities that a randomly selected seedling
was damaged, given each level of cover.
The conditional probabilities are:
P(D|no cover)=60/211=0.28.
P(D|cover<1/3)=76/234=0.3248.
P(D|cover 1/3 to 2/3)= 44/221=0.1991.
P(D|cover > 2/3)= 29/205=0.1415.
(c) Are cover and damage independent or not?
Cover and damage are not independent. The probability of getting
damaged decreases with more covers. If they were independent,
the conditional probabilities would be same and equal to P (D).
(d) Find the conditional probability that a randomly selected seedling
had no cover, given that it was damaged.
60
P (nocover|damaged) = 209
= 0.287.
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3. (20 pts pts) It is known the probability of getting a head in a toss of a
biased coin is 0.7. You are about to toss this coin 6 times.
(a) Find the probability of getting 6 heads in 6 tosses.
P (HHHHHH) = 0.76 = 0.1176.
(b) Find the probability of getting 6 tails in 6 tosses.
P (T T T T T T ) = 0.36 = 0.0007.
(c) Find the probability of getting 4 heads followed by 2 tails in 6
tosses. P (HHHHT T ) = 0.74 ∗ 0.32 = 0.0216
(d) (extra credit 1 pts). Find the probability of getting exactly 4
heads in 6 tosses. P (4Hs) = 15 ∗ 0.74 ∗ 0.32 = 0.3241.
4. (20 pts) Near a certain exit of I-81, the probabilities are 0.23 and 0.24
that a truck stopped at a roadblock will have faulty brakes or badly
worn tires. Also the probability is 0.38 that a truck stopped at the
roadblock will have faulty brakes and/or badly worn ties. What is the
probability that a truck stopped here will have
(a) both faulty brakes and badly worn tires?
P( B and T)= P(B)+P(T)-P(B or T)=0.23+0.24-0.38=0.09.
(b) neither faulty brakes nor badly worn tires?
P(neither)=1-0.38=0.62.
5. (15 pts)
(a) You must choose an SRS of 20 from the 100 retail outlets in New
York that sells your company’s products. How would you label
this population in order to use the random digit table?
a). 000, 001, .... 099, 100.
b). 001, 002, .... 099, 100.
c). 1, 2, 3, ... 99, 100.
d). 01,02, 03, ... 99, 100.
answer b).
(b) A marketing class designs two videos advertising a Mercedes sports
car. They test the videos by asking fellow students to view both
(in random order) and say which makes them more likely to buy
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the car. Mercedes should be reluctant to agree that the video
favored in this study will sell more cars because
a). the study used a matched pair design instead of a completely
randomized design.
b). results from students may not generalize to the older and
richer customers who might buy a Mercedes.
c). this is an observational study, not an experiment.
answer b).
(c) A simple random sample (SRS) of size n means
a). every individual in the population has the same chance of
being selected.
b). every set of n individuals in the population has the same
chance of being selected.
c). the sample must be chosen using the random digit table.
answer b).
(d) A sample of households in a community is selected at random from
the telephone directory. In this community, 4% of households have
no telephone, 10% have only cell phones, and another 25% have
unlisted telephone numbers. The sample will certainly suffer from
a). nonresponse.
b). undercoverage.
c). false response.
answer b).
(e) A medical study compares two muscle strengthening methods.
The researchers obtained 25 pairs of twins. One person in each
pair tried method A and the other person tried method B. For
each pair, the assignment of either method is randomly decided.
This is
a). an observational study
b). a matched pairs experiments
c). a completely randomized experiment.
d). a double blind study.
answer b).
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(f) Eighty auditors in a company signed up for an experiment in which
the efficacy of two training programs was investigated. The researcher divided the 80 participants into two groups. In doing so,
she selected the first 40 on the sign- up list and assigned them to
the program A and the last 40 to program B. After one month of
training, the researcher obtained the test scores for the two groups
of participants. One major flaw of this experiment was that
a). there were not enough subjects in each group.
b). the experiment was not randomized.
c). the experiment was not double blind.
d). the experiment did not use matched pairs design.
answer b).
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