Answer to Question #139996 in Statistics and Probability for Manqoba Gebashe

Question

The time (X) needed to complete a final examination in a particular college course is normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes.

1.1) What is the probability of completing the exam in less than 72 minutes (inclusive)? (4)

1.2) What is the probability that a student will complete the exam in more than 75 minutes (inclusive) but less than 80 minutes(inclusive)? (6)

Answer 1

Let $X=$ the time needed to complete a final examination: $X\sim N(\mu, \sigma^2).$

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

Given $\mu=61, \sigma=9$

1.1

$$P(X\leq72)=P(Z\leq\dfrac{72-61}{9})=P(Z\leq\dfrac{11}{9})\approx$$

$$\approx0.889188$$

1.2

$$P(75\leq X\leq80)=P(X\leq80)-P(X<75)=$$

$$=P(Z\leq\dfrac{80-61}{9})-P(Z<\dfrac{75-61}{9})=$$

$$=P(Z\leq\dfrac{19}{9})-P(Z<\dfrac{14}{9})\approx$$

$$\approx0.9826186-0.9400931\approx0.042526$$