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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).

Solution

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.