Question

What is the probability that a randomly chosen female has height between 145 and 157 cm?

Answer 1

The heights of an adult female population are normally distributed with mean 160 cm and standard deviation 8 cm.

Let $X=$ the height of an adult female: $X\sim N(\mu, \sigma^2).$

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

Given $\mu=160\ cm, \sigma=8\ cm.$

$$P(145<X<157)=P(X<157)-P(X\leq145)$$

$$=P(Z<\dfrac{157-160}{8})-P(Z\leq\dfrac{145-160}{8})$$

$$=P(Z<-0.375)-P(Z\leq-1.875)$$

$$\approx0.35383-0.03040\approx0.3234$$

The probability that a randomly chosen female has height between 145 and 157 cm is 0.3234.