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Lecture 5
Homework
To get full credit, show your work. For lengthy computations, avoid rounding too much
until the very end. This will make your answer as accurate as possible.
1. The scores X for a Physics nal examination follow a normal distribution with = 65 and
= 10.
(a) Fill in the blanks: X e N (
,
).
(b) Find the probability that a randomly chosen score is between 70 and 80. Answer: 0.242
(c) Find the 95th percentile for this distribution of scores. Explain what this number means.
Answer: 81.449
(d) What fraction of the scores are less than 65? Answer: 1/2
(e) If 500 students took the nal exam, about how many of the 500 are expected to score below
55? Answer: 79.328 (about 79 students)
(f) A randomly chosen student scores 80. What fraction of the scores is greater than this student's
scores? Answer: about 7/100 (5.5.12)
In 1941, the distribution of batting averages for major league baseball players with a sucient
number of at-bats to qualify for the batting title was approximately normally distributed with
mean = 0.278 and standard deviation = 0.037. In 1997, the corresponding mean and standard
deviation were approximately 0.270 and 0.030, respectively.
(a) What is the probability that a randomly chosen batting average from the 1941 distribution of
batting averages exceeds 0.400? Answer: 4.88 10 4
(b) What is the probability that a randomly chosen batting average from the 1997 distribution of
batting averages exceeds 0.400? Answer: 7.3510 6
(c) Based on your answers to parts (a) and (b), in which of the two years was it easier to hit for
such a high average? (5.5.15) Answer: 1941
2.
A stock index consists of the market value of a collection of stocks. The rate of return on stock
indices follows an approximately normal distribution. In the period of 1900-1999, the Standard &
Poor's 500 index, abbreviated S & P 500, has produced an average annual return of approximately
11:5%, with a standard deviation of about 18%.
(a) In what range do the middle 50% of all stocks yearly returns lie? (As with other problems, it
may help to draw a picture.) Answer: (-0.641%,23.641%)
(b) The stock market is said to be down on any year if the return on the S & P 500 is below zero.
In what percent of years is the market down? Answer: 26.145%
(c) Investors have \a lot to cheer about" if in any one year S & P 500 index returns in excess of
20%. In what percent of years does the index gain over 20%? Answer: 31.838% (5.5.17)
3.
You are planning to move to a certain part of the country but are concerned with the temperature
during the summer months. Literature sent to you by the Chamber of Commerce indicates that
during the summer months, the daily temperature, measured at noon, is normally distributed with
mean = 80 degrees Fahrenheit and standard deviation = 4 degrees Fahrenheit.
(a) If summer lasts exactly 90 days, how many summer days are expected to have a temperature
exceeding 90 degrees Fahrenheit? Answer: 0.558, about 1
(b) Find the 90th percentile of the distribution of daily temperatures and explain what this number
means. Answer: 85.126 degrees
(c) How many summer days are expected to have a temperature below 80 degrees Fahrenheit?
Answer: 45 days (5.5.20)
4.
The quality control exercise in your notes is simply Example 12 on 5-29 in your book. Looking
at the illustration on page 5-30, did we underestimate or overestimate the actual probability? The
answer is given on page 5-29.
5.
Twenty ve percent of the American work force works in excess of 50 hours per week. If a
sample of one hundred American workers is taken, what is the probability that thirty or more
work over 50 hours per week? Answer: 0.124 (5.6.3)
6.
Reminder: Don't use the continuity correction for this problem or any other binomial random
variable problems.
1
Americans are seldom enthusiastic about excessive government regulations, but when it comes to
junk mail in the Internet, about 75% of Internet users favor some form of governmental guidelines.
If a sample of 100 Internet users is taken, what is the probability that at least eighty will favor
some form of government regulations? Answer: 0.124 (5.6.4)
7.
8. Let X e B (50; 0:3). We wish to nd the probability of having between 12 and 25 successes
(inclusive). Answer: 0.822 (5.6.5)
The presence of bacteria in a urine sample is often a strong indication of kidney infection. It
is known that about 5% of the population have bacteria in their urine. Suppose a random sample
of 800 people is taken. What is the probability that at least 60 will have bacteria in the urine?
Answer: 5.8910 4 (5.6.6)
9.
Of the PhD's in mathematics/statistics awarded to US citizens, about 35% are earned by
women. Suppose a sample of 400 mathematics/statistics PhD's is taken.
(a) How many in the sample are expected to be men? Answer: 260
(b) What is the probability that at least 140 are women? Answer: 1/2
(c) What is the probability that fewer than 200 are men? Answer: about 0 (5.6.9)
10.
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