Question

The failure rate in a statistics class is 20%. In a class of 30 students, find the probability that exactly five students will fail. Use the normal distribution to approximate the binomial distribution.

Answer 1

Solution:

Given that,

P = 0.20

1 - P = 0.80

n = 30

Here, BIN ( n , P ) that is , BIN (30 , 0.20)

then,

n*p = 30 * 0.20 = 6 > 5

n(1- P) = 30 * 0.80 = 24 > 5

According to normal approximation binomial,

X Normal

Mean = = n*P = 6

Standard deviation = =n*p*(1-p) = 30 * 0.20 * 0.80 = 4.8

We using continuity correction factor

P(X = a) = P( a - 0.5 < X < a + 0.5)

P(4.5 < x < 5.5) = P((4.5 - 6)/ 4.8) < (x - ) /  < (5.5 - 6) / 4.8) )

= P(-0.68 < z < -0.23)

= P(z < -0.23) - P(z < -0.68)

= 0.4090 - 0.2483

= 0.1608

The probability that exactly five students will fail is 0.1608