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The mean (which is the same thing as expected value) of this distribution is 2 and that the standard deviation is the square root of 2 (or 1.4121...)
a) Multiply x by (P(x) and
sum. b) .9545; c) .2;
d) We do this table:
0,0 0,1 0,2 0,3 0,4
1,0 1,1 1,2 1,3 1,4
2,0 2,1 2,2 2,3 2,4
3,0 3,1 3,2 3,3 3,4
4,0 4,1 4,2 4,3 4,4
There are twenty-five possible outcomes. The diagonals from lower left to upper right each have the same sum and average. Four have an average of 1.5, five average 2, and four average 2.5, so 13 of the 25 are in that range, or a probability of .52.
3. A Christmas tree farm planted 100 acres of trees five years ago. These trees now have an average height of 80 inches with a standard deviation of 5 inches.
a) .1587; b) much less than 0.01%; c) .1585; d) .9545.
4. Early in the last century, sports fishermen introduced non-native species of trout into most of the streams in the West and these non-native species eliminated the native species. In recent years agencies of the government have tried to undo what was done by extirpating the introduced species and re-introducing the native species into some of these streams.
Suppose that a year after reintroduction, the Bonneville cutthroat trout have an average length of 4 inches with a standard deviation of .4 inches.
a) .4013; b) 15.87; c) .0062, which is a result that is very unlikely if the goal is in fact met.
5. A biologist is coring bristlecone pine trees on Wheeler Peak in Eastern Nevada to determine their age. Suppose that the average age of the trees in this area is 2500 years with a standard deviation of 500 years.
a) z = 1.65 = (x - 2500)/500;
x = 2500 +825 = 3325
b) z = 1.65 = (x - 2500)/100; x = 2500 +165 =
2665
6. Suppose that the average age of people who die is 86 and the standard deviation is 10 years.
a) .3829; b) .6827
7. A box with a million tickets has an average of 60 with a standard deviation of 5. If a sample of 400 draws is made from this box, we expect an average of ______________ give or take ____________or so. The probability that the average would be less than 59 is ________________________.
8. Suppose that you are fairly sure that you are dealing with a random variable that is normally distributed with a mean of 5 and a standard deviation of 1. Is it likely that such a distribution would yield a random sample of these numbers: {4, 5, 5, 5, 6, 6, 7, 7, 8, 8}? Explain carefully how you get your answer.
The sample mean is 6.1. We expect on average an error of one divided by the square root of 10, or about .316. We are more than three standard errors away from where we expect to be, which is something that is quite unusual, so we should doubt that we are dealing with a N(5,1) distribution.
9. The expected value of the random digits (zero to nine) is 4.5, their standard deviation is 2.67, and they are distributed uniformly.
a) The expected value or
sample mean is 4.5 give or take about .84.
b) 65.45%; c) 93.60%
10. Suppose that you had a million dollars to invest but you had to invest it in one of two options. In option A you have a 40% chance of doubling you money in a year, a 40% chance of just getting back your million dollars, and a 20% chance of losing it all. With option B you are sure that in one year you will receive $1.1 million.
a) Option A's expected value:
.4($200000) + $.4(1000000) -.2($1000000) = $1200000; Option
B's expected value is 100,000 less at $1100000.
b) Most people would purchase Option B because there is
great risk in option A. c) and d) However, if we could buy
little bits of many different Option As, the risk of losing
money is greatly decreased. Some of them will fail, but most
of them will succeed. If we can buy enough of Option C, we
will get the return of Option A with a risk similar to
Option B. It is on this principle that insurance and the
gambling industry is built. It is sometimes called the law
of large numbers. Life is full of small risks. Sometimes you
lose, but the cost of insuring against them is greater than
the expected loss.