Answer to Question #201108 in Statistics and Probability for Jay

Question

The weights of the Grade 11 students at the University of Santo Tomas are normally distributed with a mean of 82 pounds and a standard deviation of 7 pounds.

a. What is the probability that a randomly grade 11 student in the University of Santo Tomas is above 90 pounds?

b. What is the probability that a randomly grade 11 student in the University of Santo Tomas is below 80 pounds?

c. What is the probability that a randomly grade 11 student in the University of Santo Tomas is between 75 pounds and 85 pounds?

d. If there are 186 students at the University of Santo Tomas, how many students weigh below 80 pounds?

Answer 1

Let $=$ the weight of the Grade 11 student: $X\sim N(\mu, \sigma^2).$

Given $\mu=$ 82 pounds, $\sigma=$ 7 pounds.

a.

$$P(X>90)=1-P(X\leq 90)$$

$$=1-P(Z\leq \dfrac{90-82}{7})\approx1-P(Z\leq1.142857 )$$

$$\approx0.126549$$

b.

$$P(X<80)=P(Z< \dfrac{80-82}{7})$$

$$\approx P(Z<-0.285714 )\approx0.387549$$

c.

$$P(75<X<85)=P(X<85)-P(X\leq 75)$$

$$=P(Z< \dfrac{85-82}{7})-P(Z\leq \dfrac{75-82}{7})$$

$$\approx P(Z\leq0.428571 )-P(Z\leq-1 )$$

$$\approx0.6658824-0.1586553\approx0.507227$$

d.

$$0.387549\cdot186=72$$

72 students.