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MATH 3070: Worksheet No.4
NAME:
1. Let X be a normal random variable with mean 100 and standard deviation 8. Find the following
probability:
(a) P(X > 100)
(b) P(X < 115)
(c) P(88 < X < 120)
2. The College Board Exams are administered each year to many thousands of high school students,
and are scored so as to yield a mean of 500 and a standard deviation of 100. The distribution of the
scores is close to a normal distribution. What percentage of the scores can be expected to satisfy
each of the following conditions?
(a) Greater than 700.
(b) Less than 450.
(c) Between 450 and 600.
(d) An exclusive club wishes to invite those scoring in the top 10% on the College Board Exam
to join. What score is required to be invited to join the club?
(e) What score separates the top 60% of the population from the bottom 40%?
3. The level of a particular pollutant, nitrogen oxide, in the exhaust of automobiles has approximately a
normal distribution with a mean level of 2.1 g/m (grams per mile) and a standard deviation of 0.3
g/m.
(a) If the EPA mandates that a nitrogen oxide level of 2.7 g/m cannot be exceeded, what
proportion of automobiles would be in violation of the mandate?
(b) What nitrogen oxide level does 25% of automobiles exceed?
4. There is a considerable variation in blood pressure during a day. The systolic blood pressure reading
of a patient of high blood pressure has a normal distribution with a mean of 160 mm mercury and a
standard deviation of 20 mm. If the systolic reading exceed 150 mm, a patient will be considered to
have high blood pressure. Then answer the following questions:
(a) What is the probability that a single measurement will fail to detect that the patient has high
blood pressure?
(b) If five measurements are taken at various times during the day, what is the probability that
the average blood pressure reading will be less than 150, and hence fail to indicate that the
patient has a high blood pressure problem?
5. The rate of return of investment in one company has a normal distribution with mean 4% and
standard deviation 2%. The rate less than 2% is not considered as a good investment.
(a) What is the probability that the investment in a particular company turns out not to be a
good investment?
(b) Suppose that ten different companies are invested, and that the average rate of return is
acquired. What is the probability that this investment strategy does not become a good one?
6. Suppose that you scored 650 on SAT in 2000, and 30 on ACT in 2001. The SAT exam in 2000 had
mean 500 and standard deviation 100, and the ACT had mean 21 and standard deviation 4.7. How
can you compare these scores? Can you say you did it better in 2001?