Question

find the probability that a data value picked at random from a normally distributed population will have a standard score that corresponds to the following Less than 3.00

Answer 1

`\mathbb P(Z<3)=\mathbb P(Z<-3)+\mathbb P(|Z|<3)`

The empirical rule says that approximately 99.7% of any normal distributions falls within three standard deviations of the mean, which leaves 0.3% outside that range, and due to the distribution's symmetry, 0.15% to either side. This means


`\mathbb P(Z<-3)=0.0015`

`\mathbb P(|Z|<3)=0.997`

So


`\mathbb P(Z<3)\approx0.0015+0.997=0.9985`

or about 99.9%.