Answer to Question #103573 in Statistics and Probability for Maow Abdullahi

Question

Q2a).. The income of families in a slum have a normal distribution with a mean of $ 52 per year and a standard deviation of $ 5 . find (10mks)
i)Percentage of families with an income of more than $50 per year
ii). Percentage of families with an income of between $ 48 and $ 54 per year
iii) .In a sample of 500 families in this slum, how many have an income of more than $ 50 per year?

Answer 1

Let X=the income of families in a slum. $X\sim N(\mu, \sigma^2)$

Then

$Z={X-\mu \over \sigma}\sim N(0, 1)$

Given that $\mu=52, \sigma=5$

i)

$P(X>50)=1-P(X\leq50)=$ $=1-P(Z\leq{50-52 \over 5})=1-P(Z\leq-0.4)\approx$ $\approx1-0.344578\approx0.655422$

$65.54\%$

ii)

$P(48<x<54)=P(X<54)-P(X\leq48)=$ $=P(Z<0.4)-P(Z\leq-0.8)\approx$ $\approx0.655422-0.211855=0.443567$

$44.36\%$

iii)

Given that $n=500$

$P(X>50)\approx0.655422$

$500(0.655422)\approx328$

Approximately 328 families in a sample of 500 families in this slum have an income of more than $ 50 per year.