Question

According to a government study, 8% of all children live in a household that has an income below the poverty level. If a random sample of 20 children is selected:

a) what is the probability that 5 or more live in poverty?

b) what is the probability that 4 live in poverty?

c) what is the expected number (mean) that live in poverty? What is the variance? What is the standard deviation?

Answer 1

Let p denote the probability of selected child belong to household below poverty line.

p = 8% or 0.08

q = probability that child selected doesn't belong to below poverty line household

q = 1 - p

q = 1 - 0.08

q = 0.92

We need to use the binomial distribution to solve this question.

The formula for binomial distribution is given as,

P(X=r) = ncr × p^r × q^1-r

Here r = number of success

ncr = n!/(n-r)! r!

n = total number number of trial

p = probability of selecting a child belonging to below poverty line

q = probability of child selected being not belonging to below poverty line.

(a) Now we need to calculate the probability that at 5 children belongs to below poverty households.

P (X >=5)

And this can be calculated as,

= 1 -[P(X=0)+ P(X=1) + P(X=2) + P(r=3) + P(r=4)]

= 1 - [ 20C0 (0.08)^0(0.92)^20 + 20C1×(0.08)^1 × (0.92)^19+ 20C2×(0.08)^2(0.92)^18 + 20C3×(0.08)^3(0.92)^17 + 20C4×(0.08)^4(0.92)^16]

= 1 - [ 0.18×1 + 20×0.08×0.20 + 190×0.0064×0.22 + 1140×0.00051×0.24 + 4845×0.0000040×0.26]

= 1 - [0.18 + 0.32 + 0.26 + 0.13 + 0.05]

= 1 - 0.94

P(X >= 5) = 0.06

So the probability of getting 5 or more children belonging to below poverty line income household is equal to 0.06.

(b) Now we need to calculate the probability that 4 children live in poverty.

P(X=4) = 20C4 × (0.08)^4 × (0.92)^16

If you notice we have already calculated this term in above part,

P(X=4) = 4845×0.0000040×0.26

P(X=4) = 0.05

So the probability that 4 children lives in poverty is equal to 0.05.

(c) Now we need to calculate the expected number of children or mean of Childers who lives in poverty.

So the mean of binomial distribution is given as,

= np

Here n = 20 that is total number of children in the sample

p = 0.08 probability that child belongs to below poverty level income house or lives in poverty.

Putting in the values we get,

= 20 × 0.08

= 1.6

So the expected number of mean of children living in poverty is equal to 1.6.

Similarly the formula for standard deviation is given as

= (npq)^1/2

Here n = 20

p = 0.08

q = 0.92

Putting in the values we get,

= (20×0.08×0.92)^1/2

= 1.21

So the standard deviation is equal to 1.21.

Note: I know the maths looks very tedious but please take the calculations on faith and just focus on the formulas and concepts and you'll be able to it.

I hope I was able to help you, thank you.