Stat 330 Homework 7: Specific Continuous Distributions Spring
2009
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Convenience Sample
The average time between customers arriving at a store is 10 minutes. The inter-arrival times are independent and exponentially distributed. Every fifth customer receives a questionnaire. (Specifically, only the
5th , 10th , 15th , ... customers receive questionnaires.)
(a.) If the store opens at 8:00 am, at what time does the store owner expect to give the first questionnaire?
On one slow morning, no customers have arrived by 9:00 am. Now, what time does the owner expect
to give the first questionnaire?
(b.) At 1pm, the store owner takes a lunch break. How many customers does the owner expect to arrive
before the lunch break? How many questionnaires does the store owner expect to distribute before the
lunch break?
(c.) One morning, the store owner realizes that he only has 3 questionnaires left. He will make more
questionnaires during his lunch break. What is the probability that the store owner runs out of
questionnaires before his lunch break?
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Playing CDs
Assume that the playing time of jazz CDs has a normal distribution with a mean of 52 minutes and a
standard deviation of 7 minutes.
(a) If you select a random jazz CD from the sale’s stack has a playing time of less than 45 minutes?
(b) What percentage of jazz CDs play between 45 and 59 minutes?
(c) Determine the percentage of jazz CDs that fall within a range of ±2σ around the mean playing time.
(d) Determine the percentage of jazz CDs that fall within a range of ±3σ around the mean playing time.
(e) What is the relative frequency of jazz CDs with playing time X exactly 45 minutes?
(f) What is the relative frequency of jazz CDs with playing time over 1 hour?
(g) Given that a CD a randomly chosen CD has been playing for 59 minutes, what is the conditional
probability that the CD continuis to play for at least 14 more minutes?
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NY Stock Exchange
For 1998 as a whole, the mean return of all common stocks listed on the New York Stock Exchange (NYSE)
was µ = 16% and standard deviation σ = 26%. Assume that the distribution of returns is roughly normal.
(a) What % of stocks lost money?
(b) Suppose we create a portfolio of 8 stocks by randomly selecting stocks from the NYSE and investing
equal amounts of money in each stock. What are the mean and standard deviation of the sample mean
of the returns for these 8 stocks?
(c) What is the probability the portfolio loses money? Explain the difference between this result and that
of part (a)?
(d) The probability is 0.05 that a portfolio constructed this way has a return of more than
?
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Apple-Tree Farm
Able and Baker are both apple-tree farmers (they grow apple trees). Assume, that apple trees grow according
to a normal distribution.
On the Able Farm, trees grow with a mean of 1m per year and a standard deviation of 25 cm. Baker manages
to get an average growth of 1.1m per year with a standard deviation of 35cm.
For the following questions state each time which random variable you use and what distribution assumption
you make.
(a) What is the probability for a tree on the Able Farm to grow between 0.7m and 0.9m in a year? what
if the tree was on the Baker Farm?
(b) Assume, you’ve got two trees. One from Able and one from Baker. What can you say about the
difference D in their heights? What is the distribution of D?
(c) On average, trees from the Baker farm will grow more than Able’s trees. But what is the exact
probability that a Baker tree has grown more than an Able tree in one year?
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