Answer to Question #168878 in Statistics and Probability for Hazel

Question

How to calculate normal distribution between 60 and 70 with a mean 65 and SD of 5?

Answer 1

Let $X\sim N(\mu, \sigma^2).$ Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)$

Given $\mu=65, \sigma=5$

$$P(60\leq X\leq70)=P(X\leq 70)-P(X<60)$$

$$=P(Z\leq \dfrac{70-65}{5})-P(Z<\dfrac{60-65}{5})$$

$$=P(Z\leq 1)-P(Z<-1)$$

$$\approx 0.841345-0.15866\approx0.6827$$

The probability is 0.6827.