Homework Sourse3 The probability that an amateur archer hits a target 30 fe
3. The probability that an amateur archer hits a target 30 feet away is 20%. Suppose we take a sample of 12 archers.
a. What is the expected number of archers that will hit the target?
b. What is the standard deviation?
c. What is the probability that exactly 3 archers hit the target?
d. What is the probability that more than 4 archers hit the target?
Suppose we go back to the example from question 3. Suppose the probability remains at 20% but instead we take a sample of 100 archers.
a. What is the standard error of the proportion?
b. What is the probability that the proportion of archers who hit the target will be less than 10%?
Solution
3.
a)
E(x) = n p = 12*0.20 = 2.4 [answer]
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b)
s(x) = sqrt(n p (1-p)) = sqrt(12*0.2*(1-0.2)) = 1.385640646 [answer]
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c)
Note that the probability of x successes out of n trials is
P(n, x) = nCx p^x (1 - p)^(n - x)
where
n = number of trials = 12
p = the probability of a success = 0.2
x = the number of successes = 3
Thus, the probability is
P ( 3 ) = 0.236223201 [answer]
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d)
Note that P(more than x) = 1 - P(at most x).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 12
p = the probability of a success = 0.2
x = our critical value of successes = 4
Then the cumulative probability of P(at most x) from a table/technology is
P(at most 4 ) = 0.9274445
Thus, the probability of at least 5 successes is
P(more than 4 ) = 0.0725555 [answer]
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