A recent survey reported that 63% of 18 to 29-year-olds in a certain country own

Question

A recent survey reported that 63% of 18 to 29-year-olds in a certain country own tablets. Using the binomial distribution, complete parts (a) through (e) below.

a. What is the probability that in the next six 18- to 29-year-olds surveyed, four will own a tablet?

b. What is the probability that in the next six 18- to 29-year-olds surveyed, all six will own a tablet?

c. What is the probability that in the next six 18- to 29-year-olds surveyed, at least four will own a tablet?

d. What are the mean and standard deviation of the number of 18- to 29-year-olds who will own a tablet in a survey of six?

e. What assumptions do you need to make in (a) through (c)? Select all that apply.

-The outcome of any observation is independent of the outcome of any other observation.

-The probability of an observation being classified as the event of interest, pi, is constant from observation to observation.

-The outcome of any observation is dependent of the outcome of any other observation.

-Each observation is classified into one of two mutually exclusive and collectively exhaustive categories.

Explanation / Answer

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a.
P( X = 4 ) = ( 6 4 ) * ( 0.63^4) * ( 1 - 0.63 )^2
= 0.3235

b.
P( X = 6 ) = ( 6 6 ) * ( 0.63^6) * ( 1 - 0.63 )^0
= 0.0625

c.
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 6 3 ) * 0.63^3 * ( 1- 0.63 ) ^3 + ( 6 2 ) * 0.63^2 * ( 1- 0.63 ) ^4 + ( 6 1 ) * 0.63^1 * ( 1- 0.63 ) ^5 + ( 6 0 ) * 0.63^0 * ( 1- 0.63 ) ^6
= 0.3937
P( X > = 4 ) = 1 - P( X < 4) = 0.6063

d.
Standard Deviation ( npq )= 6*0.63*0.37 = 1.1826

e.
-The outcome of any observation is independent of the outcome of any other observation

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