Question

It has been stated that 12% of the general population donate their time and energy to working on community projects. Suppose 3 people have been randomly selected from a community and asked if they donated their time and energy to community projects.

a)      Calculate the values of P(X=k) for k=0,1,2,3.

b)       Graph P(X=k) .

c)      Find the probability that at least 1 of those 3 do donate their time and energy.

d)      Find the mean and the variance of this probability distribution and give their statistical meanings. Locate the mean on your graph.

e)      Find probability of the event (X is more than µ + 2σ).

Answer 1

Here X= Number of people donated their time and energy to community projects out of 3.

p= proportion of  the general population donating their time and energy to working on community projects, which is given as .12

Then X~Binomial(3,.12) , that is,

a) Thus we calculate the values of P(X=k) as

k 0 1 2 3

P(X=k):   0.681472 0.278784 0.038016 0.001728

b)

c) P( at least 1 of those 3 do donate their time and energy)=P(X>=1)=1-P(X=0)=1-0.681472=0.318528

d) µ=E(X)=0* 0.681472+1* 0.278784 +2* 0.038016+3*0.0017280=.36

Thus out of a large group of 100 such people, 36 are expected to donate their time and energy to community projects.

E(X2)=0* 0.681472+12* 0.278784 +22* 0.038016+32*0.0017280=0.4464

Var(X)=E(X2)-E2(X)= 0.3168

e) σ=SD=square root of Var(X)= 0.5628499

P(X>µ + 2σ)=P(X>.36+2*0.5628499)=P(X> 1.4857)=P(X>=2)=P(X=2)+P(X=3)=0.038016 + 0.001728=0.039744 as X takes only integer values.

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