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**Unformatted text preview: **MATH 218 FINAL EXAM May 7. 2001 Directions Every numerical answer should be simpliﬁed to a fraction or a decimal. You must show your work and justify your
methods to obtain full credit. Use the continuity correction whenever appropriate. If you can’t do one part of a.
problem but need that answer later, guess an answer and use that guess for the later part. The exam is worth a total of 200 points. 1. ([5 points) Consider the table below of 100 stocks classiﬁed according to type and performance over the last ye ar. Earnings > 10%
0 < Earnings S 10%
Earnings S 0
Total (a) (5 points) Find the probability that a randomly selected technology stock earned more than 10%. (b) ('5 points) A stock is selected at random. Are the events {Is a technology stock} and {Earned > 10%}
independent? (c) (5 points) Find the probability that an investor who randomly selected 4 stocks at the beginning of last
year had exactly one earn > 10%, and three earn between 0 and 10%. 2. (15 points) The Payless Insurance Company has three types of customers; high risk. medium risk and low risk.
20% of its customers are high risk, 30% are medium risk and 50% are low risk. 14.8% of all their customers had
at least one accident this last year but the lower risk customers were less prone to accidents. Indeed. 16% of
their medium risk customers and only 10% of their low risk customers had at least one accident this last year. (a) [-5 points) Draw a tree diagram that describes this situation filling in as much information as you can. (Hint:
start the tree with the events related to the three types of customers]. (b) (5 points) A randomly selected customer had an accident this last year. What is the probability that he/she
is a medium risk customer? (c) {5 points) A high risk customer is selected at random. What is the probability that he/she had at least one
accident this last year? 3. (10 points) The scores on a certain exam were normally distributed with a mean of 72 points and a standard
deviation of 5 points. (a) (.5 points} What proportion of students scored between 67.1 and 77.2 points? b 5 >oiuts The to J 15% of students are to receive an "A”. What is the lowest score to fret an ”A” on that
l l l i I 0 exam? 4. (2.5 points) The Poisson casino in Las Vegas is considering a new gambling game where a single fair die is rolled
and the pay-out is +3 (i.e. gambler wins $3} if any of the numbers 4, 5, or b‘ are rolled, 75 (i.e. gambler loses
85) if ] or ‘2 is rolled and, 0 if a 3 is rolled. (a) (10 points) Let X denote the amount won when the game is played one time. Find the expected value and
standard deviation of X. (Hint: You may ﬁnd it helpful to ﬁrst ﬁll in the blanks below in the probability distribution table ofX.)
PlX : 7’] —_—
(b) {5 points) if the gambler plays the game twice: what is the probability that the total winnings are greater
than 0'? (c) [10 points) If the. gambler plays the game 100 times, ﬁnd the approximate probability that the total winnings
are greater than 0. :3. (1:3 points) The waiting time. T1 in minutes. for a train at a particular station is a random variable with 10. probability density function _ c(1—U.lf.) if 0 g t g 10.
I”) _ l 0 otherwise. The graph of f is shown below. (a) (5 points) Find the value of c.
(b) (5 points) Find the probability that a randomly selected passenger waits less than 5 minutes. (c) (5 points) You have been waiting for the train for 5 minutes. What is the probability that you will have to
wait for at most 3 more minutes? (15 points) 10% of all people contacted by a life insurance company buy insurance from that company. Using
an appropriate approximation ﬁnd the probability that out of 600 people contacted by the company 55 or fewer
buy life insurance. You may assume that the 600 people form a simple random sample. (15 points) A downtown hotel ﬁlls an average of 71 rooms per weekend with a standard deviation of 15 rooms.
Suppose that the distribution of the number of rooms ﬁlled is (approximately) normal. (a) (5 points) The hotel schedules adequate staff to handle a response of 80 rooms. What is the probability
that additional staff will be needed on one particular weekend? (b) (10 points) The hotel runs a special promotion to try to ﬁll rooms that aren’t usually occupied on weekends.
The promotion manager reviews the weekend demand over a 10—week period. If the average demand over
this period exceeds 80, the manager will increase the scheduled staff. Suppose the promotion is not successful
and the distribution of the number of rooms ﬁlled remains unchanged. What is the probability that the
average demand over the 10—week period will exceed 80‘? You may regard the 10-week period as a random
sample. (25 points) A real estate agent would like to enter the Malibu luxury housing market and is interested in assessing
the mean value of homes in this category. To do so, she randomly selects four homes. These homes are valued
at 5,6.7. and 4 million dollars. Assume that the distribution of values is normal. (a) (10 points) Provide point estimates for the mean and standard deviation of the value of Malibu luxury
homes. (b) (10 points} Determine a 95% conﬁdence interval for the mean value of such homes. (c) (5 points) Determine how many additional observations the agent needs in order to estimate the mean value
of Malibu luxury homes to within :5 million dollars with 95% conﬁdence. (15 points) The NIH is trying to assess the prevalence of Romulan Fever. They randomly select 100 members
of the population and observe them for one year. Six of the 100 people observed develop Romulan Fever. Find
a 95% conﬁdence interval for p. the true proportion of the whole population that develops Romulan Fever in a one year period. (20 points] Before last year’s crisis in the Japanese stock market. the average foreign investment per company
among all Japanese technology companies was 250 million Yen. The Finance minister of Japan wants to know
if this has declined. However. it is intrusive and difﬁcult to obtain information from all companies mid—year. so
he has his assistant consider a random sample of 15 technology companies and perform a hypothesis test. (a) (-5 points) State the appropriate null and alternative hypotheses. Be sure to explain your reasoning. (b) (10 points) Choose a test statistic and rejection rule to test the null hypothesis at. the. 5% signiﬁcance
level, using the. random sample of 15 technology companies. You may assume that investments in Japanese
technology companies are normally distributed. (c) (5 points) After some armtwisting the assistant determines that the current average foreign investment
for the 15 companies selected is 238.5 million You with a sample stand lard deviation 3 c _‘20 1111111011 Yen Should the null hypothesis be rejected? 1111110111 your reasoning. 1.(20 points) Charles and David are playing a game that invollves rolling a die. Iliev suspect that the die might be weighted so that the chance of rolling a t) is not equal 1.330 — Thev decide to each perform an hypothesis test 01H” :p: % versus Ha :p95% (a) 10 points] Charles rolls the die 50 times and gets 13 sixes. Calculate his P-value. JpOll‘ltS) )David rolls the die a much greater number of times but also gets 26% of the 1011s to be sixes Is
is P- value greater than less than. 01' equal to Charles P— value‘.7 Explain. 1 (b) [5 points) Should Charles reject H0 at the 10% signiﬁcance level” Explain.
(C1 (5
11 12. {10 points) (a) (5 points) A certain disease occurs in 2% of the population. it is suspected that higher levels of chromium in
the water may increase the rate of occurrence of this disease. One hundred different locations are identiﬁed
that have higher levels of chromium in the water. At each of these locations 3 test is clone of H0 : p 2 0.02
versus Ha : p > 0.02 Where p represents the proportion of the population in that location that acquire
the disease. The tests are performed at the 5% level. Suppose that, in actual fact, the higher levels of
chromium in the water have no effect on the rate. of occurrence of this disease and that p = 11.02 in each
of these locations. In how many of the locations would we expect the tests to reveal incorrectly that the
alternative hypothesis. 1) > 0.02, be true? Explain. (b) (5 points) Alex and Elizabeth each perform a test of H0 : ,u S 30 versus Ha : ,u > 30. Alex uses a sample
of size 50 and Elizabeth uses a sample of size 200. Remarkably, they get the same P—value. Who has more
compelling evidence for HE, or is their evidence equally compelling? Explain. ...

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