Homework SourseSuppose that 40 of Iowans are in favor of a presidential can
Suppose that 40% of Iowans are in favor of a presidential candidate. A pollster selects a random sample of 10 Iowans and asks them about their opinions about the candidate. Let X be the number of people in the sample who are in favor of the candidate.
(a) What is the sample space of X?
(b) What is the distribution of X? 1
(c) What is the probability that 5 people in the sample are in favor of the candidate?
(d) What is the probability that 3 to 5 people in the sample are in favor of the candidate?
(e) What is the probability that at least 6 of the people in the sample are in favor of the candidate?
Solution
a)
X = {0,1,2,3,4,5,6,7,8,9,10}
b)
P(x) follows the binomial distribution,
P(x) = nCx 0.40^x 0.60^(n - x) [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 = 10
p = the probability of a success = 0.4
x = the number of successes = 5
Thus, the probability is
P ( 5 ) = 0.200658125 [ANSWER]
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d)
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 = 10
p = the probability of a success = 0.4
x = the number of successes = 3
Thus, the probability is
P ( 3 ) = 0.214990848
Similarly,
P ( 4 ) = 0.250822656
P ( 5 ) = 0.200658125
Adding these probabilities,
P(3 to 5) = 0.666471629 [ANSWER]
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E)
Note that P(at least 6) = 1 - P(at most 5).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 10
p = the probability of a success = 0.4
x = our critical value of successes = 6
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 5 ) = 0.833761382
Thus, the probability of at least 6 successes is
P(at least 6 ) = 0.166238618 [ANSWER]
