Question

Five hundred children participated in a field demonstration. Their heights averaged is 110 cm with a standard deviation of 6 cm. What is the probability that the height of a child, picked at random, has a between 110 cm and 122 cm?

Answer 1

Let $X$ be a random variable denotes the height of the students.

Then $X$ is normally distributed with mean $110$ cm and standard deviation $6$ cm: $X\sim N(110, 6^2).$

$$P(110<X<122)=P(X<122)-P(X\leq110)$$

$$=P(Z<\dfrac{122-110}{6})-P(Z\leq\dfrac{110-110}{6})$$

$$=P(Z<2)-P(Z\leq0)$$

$$\approx0.97725-0.5=0.47725$$

$P(110<X<122)=0.47725$