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ID: 3441962 • Letter: H
Question
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Question
A recent survey reported that 22 % of adults 55 and over own a smart phone. Using the binomial distribution...?
what is the probability that in the next 6 adults 55 older and older surveyed..
A. four will own a smart phone?
B. all six will own a smart phone?
C. at least four will own a smartphone?
D. What are the mean and standard deviation of the number of adults 55 and older who will own a smartphone in a survey of six?
what assumptions do you need to make in (A) through (C).
Explanation / Answer
A.
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 = 6
p = the probability of a success = 0.22
x = the number of successes = 4
Thus, the probability is
P ( 4 ) = 0.021378203 [answer]
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b)
P(All six) = 0.22^6 = 0.00011338 [answer]
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c)
Note that P(at least x) = 1 - P(at most x - 1).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 6
p = the probability of a success = 0.22
x = our critical value of successes = 4
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 3 ) = 0.976096518
Thus, the probability of at least 4 successes is
P(at least 4 ) = 0.023903482 [ANSWER]
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d)
Mean = n p = 6*0.22 = 1.32 [answer]
standard deviation = sqrt[n p (1 - p) = sqrt(6*0.22*(1-0.22)) = 1.014692072 [answer]
We assumed that each adult's ownership of smartphone is independent from all the other adults in the sample, and that the probability of ownership is constant at 0.22.
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