Question

An instructor has 1000 exams that will be graded in sequence. The times required to grade exams are iid random variables with M=20 minutes and o=4 minutes. Find the probability that the instructor will grade at least 25 exams in the first 450 minutes of work.

Answer 1

Solution:

Find the probability that instructor will grade atleast 25 exams in first 450 minutes.

Here we have to use central limit theorem

Central Limit theorem states that

Let be iid each with mean $\mu$ and variance $\sigma^2$ then the distribution of $\frac{X_1+X_2+X_3+...+X_n-n\mu}{\sigma \sqrt{n}}$ tends to standard normal z as n tends to infinity that is

$P(X_1+X_2+X_3+...+X_n-n\mu\leq b) \approx P(z\leq b)=\phi(b)$

so, probability that instructor grade atleast 25 exams in first 450 minutes is 0.006.

we have -→? time Solution : consider x; variable, where x; is yerga instructor takes to grade i exams So, time taken by instructor to grade 25 exams is X = X, + X2+ X3+...+X25 so to find P(x < 450) Here, use the central Limit Theorem P (x < 450) = Р a X-nu orn Given, u = 20 minutes and o= 4 minutes so, nel = 25(20) = 500 [n is number of exams n=25] oth = 4(125) = 4(5)= 20 plug values in O, it gives 450-500 P(x (450) = P(* 25° ( 450-nu 20 P(Z <-25) 1-002:5) 11 - 0.006