Answer to Question #97572 in Statistics and Probability for Juliet Beglaryan

Question

6. Exam scores for a large introductory statistics class follow an approximate normal distribution with an average score of 56 and a standard deviation of 5. The average exam score in your lab was 59.5. The 20 students in your lab sections will be considered a random sample of all students who take this class. a) What is the expected value of the average exam score of the 20 students in your lab section?b) What is the standard deviation of the distribution of the average exam score of the 20 students in your lab section?c) What is the probability that the average score of a random sample of 20 students exceeds 59.5?

Answer 1

Let $X=$ the average exam score: $X\sim N(\mu,\sigma^2)$

a) What is the expected value of the average exam score of the 20 students in your lab section?

$$E(X)=\mu=56$$

b) What is the standard deviation of the distribution of the average exam score of the 20 students in your lab section?

$$\text{standard deviation}={\sigma \over \sqrt{n}}={5 \over \sqrt{20}}={\sqrt{5} \over2}\approx1.118$$

c) What is the probability that the average score of a random sample of 20 students exceeds 59.5?

$$P(X>59.5)=P(Z>{59.5-56 \over \sqrt{5}/2})=1-P(Z\leq1.4\sqrt{5})\approx$$ $$\approx1-0.999128=0.000872$$