Answer to Question #196118 in Statistics and Probability for Ruisdael rael

Question

A school principal claims that grade 11 students have a mean grade of 86 with a standard deviation of 4 . Suppose that the distribution is approximately normal . What is the probability that a randomly selected grade will be greater than 82 but less than 90

Answer 1

Let $X=$ grade: $X\sim N(\mu,\sigma^2)$

Given $\mu=86, \sigma=4$

$$P(82<X<90)=P(X<90)-P(X\leq82)$$ $$=P(Z<\dfrac{90-86}{4})-P(Z\leq\dfrac{82-86}{4})$$ $$=P(Z<1)-P(Z\leq-1)$$ $$\approx0.84134-0.15866\approx0.6827$$