Answer to Question #286838 in Statistics and Probability for Sanford Namakando

Question

The height of students at a university taking introduction to probability are nomally distributed with mean height of 50 cm and standard deviation 10 cm. Find the probability that the height of a randomly selected standard is

(a) shorter than Wye

(b) within 5 cm of the mean

Answer 1

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

Given $\mu=50cm, \sigma=10cm$

(a)

$$P(X<W_{ye})=P(Z<\dfrac{W_{ye}-50}{10})$$

(b)

$$P(|X-\mu|<2.5)=P(47.5<X<52.5)$$

$$=P(X<52.5)-P(X\leq4705)$$

$$=P(Z<\dfrac{52.5-50}{10})-P(Z\leq\dfrac{47.5-50}{10})$$

$$=P(Z<0.25)-P(Z\leq-0.25)$$

$$\approx0.598706-0.401294$$

$$\approx0.1974$$