Question

A subdivision household collects an 20 killos of trash each week. If the standard deviation is 3.5 kilos based on the assumption that the data collected are normally distributed, find the probability that a household selected at random collect trash of:

a. lower than 16 kilos

b. greater than 22.5 kilos

c. between 14 kilos and 23.5 kilos

d. 11 kilos and 15 kilos

Answer 1

We have a normal distribution, $\mu=20, \sigma=3.5.$

Let's convert it to the standard normal distribution,

$z=\cfrac{x-\mu}{\sigma}.$

$\text{a. } \ z=\cfrac{16-20}{3.5}=-1.14;\\ P(X<16)=P(Z<-1.14)=\\ =0.1271.$

$\text{b. } \ z=\cfrac{22.5-20}{3.5}=0.71;\\ P(X>22.5)=P(Z>0.71)=\\ =1-P(Z<0.71)=\\ =1-0.7611=0.2389.$

$\text{c. } \ z_1=\cfrac{14-20}{3.5}=-1.71;\\ z_2=\cfrac{23.5-20}{3.5}=1;\\ P(14<X<23.5)=P(-1.71<Z<1)=\\ =P(Z<1)-P(Z<-1.71)=\\ =0.8413-0.0436=0.7977.$

$\text{d. } \ z_1=\cfrac{11-20}{3.5}=-2.57;\\ z_2=\cfrac{15-20}{3.5}=-1.43;\\ P(11<X<15)=P(-2.57<Z<-1.43)=\\ =P(Z<-1.43)-P(Z<-2.57)=\\ =0.0764-0.0051=0.0713 \text{ (from z-table).}$