Salary data
Salary Increase
$0
$0
$0
$0
$2,000
$2,000
$2,000
$2,000
$4,500
$4,500
$4,500
$4,500
Merit Rating
0
1
2
3
0
1
2
3
0
1
2
3
Probability
0.08
0.1
0.05
0
0.06
0.2
0.2
0.02
0.01
0.1
0.1
0.08
Means and standard deviations of weekly production (units produced) of two products
Product
Mean
Standard deviation
X
22500
2500
Y
30000
3350
64. A roulette wheel contains the numbers 0,00, and 1
to 36. If you bet $1 on a single number coming up,
you earn $35 if the number comes up and lose $1
otherwise. Find the mean and standard deviation of
your winnings on a single bet. Then find the mean
and standard deviation of your net winnings if you
make 100 bets. You can assume (realistically) that
the results of the 100 spins are independent. Finally,
provide an interval such that you are 95% sure your
net winnings from 100 bets will be inside this
interval
65. Assume that there are four equally likely states of
the economy: boom low growth, recession, and
depression. Also, assume that the percentage annual
retwn you obtain when you invest a dollar in gold or
the stock market is shown in the file P04_65.xlsx.
a. Find the covariance and correlation between the
annual return on the market and the annual retun
on gold. Interpret your answers.
b. Suppose you invest 40% of your available money
in the market and 60% of your money in gold
Determine the mean and standard deviation of the
annual retun on your portfolio.
c. Obtain your pat b answer by determining the
actual return on your portfolio in each state of the
economy and then finding the mean and variance
calculating the return on your portfolio for each
state and use the formulas for mean and variance of
a random variable. For example, in the boom state,
your portfolio eams 0.25(0.25) + 0.75(0.32).
67. Each year the employees at Zipco receive a $0, $2000,
or $4500 salary increase. They also receive a merit
rating of 0, 1, 2 or 3, with 3 indicating outstanding
performance and 0 indicating poor performance. The
joint probability distribution of salary increase and
merit rating is listed in the file P04_67.xlsx. For
example, 20% of all employees receive a $2000
increase and have a merit rating of 1. Find the corre-
lation between salary increase and merit rating. Then
interpret this correlation
68. The return on a portfolio during a period is defined by
PV
- PV
PV
bag
where PV is the portfolio value at the beginning of
period and PT od is the portfolio value at the end of
the period. Suppose there are two stocks in which you
can invest, stock 1 and stock 2. During each year there
is a 50% chance that each dollar invested in stock I will
turn into $2 and a 50% chance that each dollar invested
in stock 1 will turn into $0.50. During each year there is
end
bes
78. A manufacturing plant produces two distinct products,
A and B. The cost of producing one unit of A is $18
and that of B is $22. Assume that this plant ineus a
weekly setup cost of $24,000 regardless of the number
of units of A or B produced. The means and standard
deviations of the weekly production levels of A and B
are given in the P04_78.xlsx.
a. Assuming that the weekly production levels of A
and B are independent, find the mean and standard
deviation of this plant's total weekly production
cost. Between which two total cost figures can you
be about 68% sure that this plant's actual total
weekly production cost will fall?
b. How do your answers in part a change if you
discover that the correlation between the weekly
production levels of A and B is actually 0.29?
Explain the differences in the two sets of results.
79. The typical standard deviation of the annual retun on a
stock is 20% and the typical mean retun is about 12%.
The typical correlation between the annual retuns of two
stocks is about 0.25. Mutual finds often put an equal
percentage of their money in a given number of stocks.
By choosing a large member of stocks, they hope to
diversify away the risk involved with choosing particular
stocks. How many stocks does an investor need to own to
diversify away the risk associated with individual stocks?
To answer this question, use the above information about
"typical” stocks to determine the mean and standard
deviation for the following portfolios:
• Portfolio 1: Half your money in each of 2 stocks
Portfolio 2: 20% of your money in each of 5 stocks
Portfolio 3: 10% of your money in each of 10
stocks
Portfolio 4: 5% of your money in each of 20 stocks
Portfolio 5: 1% of your money in each of 100
stocks
What do your answers tell you about the number of
stocks a mutual fund needs to invest in to diversify
adequately?
standard deviation of their net winnings. The file gets
you started
a. Player 1 always bets on red. On each bet, he either
wins or loses what he bets. His first bet is for $10.
From then on, he bets $10 following a win, and he
doubles his bet after a loss. (This is called a martin-
gale strategy and is used frequently at casinos.) For
example, if he spins red, red, not red, and not red, his
bets are for $10, 10, $10, and $20, and he has a net
loss of $10. Or if he spins not red, not red not red,
and red, then his bets are for $10, $20, $40, and $80,
and he has a net gain of $10.
b. Player 2 always bets on black and green. On each
bet, he places $10 on black and $2 on green If red
occurs, he loses all $12. If black occurs, he wins a
net $8 ($10 gain on black $2 loss on green). If
green occurs, he wins a net $50 ($10 loss on black,
$60 gain on green).
82. Suppose the New York Yankees and Philadelphia
Phillies (two Major League Baseball teams) are playing
a best-of-three series. The first team to win two games is
the winner of the series, and the series ends as soon as
one team has won two games. The first game is played
in New York, the second game is in Philadelphia, and if
necessary the third game is in New York. The
probability that the Yankees win a game in their home
park is 0.55. The probability that the Phillies win a game
in their home park is 0.53. You can assume that the
outcomes of the games are independent
a. Find the probability that the Yankees win the series.
b. Suppose you are a Yankees fan, so you place a bet
on each game played where you win $100 if the
Yankees win the game and you lose $105 if the
Yankees lose the game. Find the distribution of your
net winnings. Then find the mean and standard
deviation of this distribution. Is this betting strategy
favorable to you?
c. Repeat part a, but assume that the games are
played in Philadelphia, then New York, then
Philadelphia. How much does this "home field
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