Answer to Question #122138 in Statistics and Probability for Chelsea

Question

At a large university, 25% of students are over 21 years old. In a sample of 400 students, calculate the probability that more than 110 of them are over 21 years old.

Answer 1

If X ~ B(n, p) and if n is large and/or p is close to ½, then X is approximately N(np, npq)

Here n is large so normal distribution can be used as an approximation to the binomial distribution.

Step 1: find mean and sd

mean = np= 400*0.25=100

sd= s=$\sqrt{np(1-p)}$ =$\sqrt{400(0.25)(1-0.25)}$ = 8.66

Step 2: since np=400*0.25= 100

nq=400*(1-0.25)= 300, both are greater than 10, we use continuity correction factor.

Step 3:

Use the continuity correction factor on the X value.

if P(X>a) use P(x>a+0.5)

Hence P(X>110)=P(X>110.5)[using continuity correction factor)

Step4: find z score

Z=(x-mean)/sd

Z= (110.5-100)/8.66

Z=1.1124

Step 5:

so P(Z>1.2124)=1-0.8873= 0.1127

so then, we conclude that P( X >110 ) approx 0.1127

P(X>110)≈0.1127, which ends the calculation of the requested probability