Answer to Question #302737 in Statistics and Probability for love

Question

How many children have heights between 98 cm and 104 cm?

Answer 1

Five hundred children participated in a field demonstration. Their heights averaged is 110 cm with a standard deviation of 6 cm. How many children belong to the upper 15% of the group?

Let $X=$ height of a child: $X\sim N(\mu, \sigma^2).$

Given $\mu=110\ cm, \sigma=6\ cm.$

$$P(98<X<104)=P(X<104)-P(X\le98)$$

$$=P(Z<\dfrac{104-110}{6})-P(Z\le\dfrac{98-110}{6})$$

$$=P(Z<-1)-P(Z\le -2)$$

$$\approx0.158655-0.022750=0.135905$$

$$0.135905\cdot 500=68$$

68 children have heights between 98 cm and 104 cm.