Question

An elementary school teacher learned that 40% of school children have at least 3 cavities. The teacher has 30 students in his class. How many students would he expect in his class to have at least 3 cavities? What is the standard deviation? Using the appropriate approximation, determine P(x>20); that is, the probability that more than 20 students in his class will have 3 cavities.

Answer 1

X ~ Binomial (n,p)

Where n = 30 , p = 0.40

E(X) = n * p

= 30 * 0.40

= 12

Expected number of students have at least 3 cavities = 12

Standard deviation = Sqrt(np(1-p))

= Sqrt( 30 * 0.40 * 0.60)

= 2.6833

Standard deviation = 2.6833

We have to calculate P( X > 20) = ?

Using normal approximation with continuity correction,

P( X > x) = P( X > x + 0.5) (Using continuity correction)

P( X > x) = P( Z > x - Mean / Standard deviation) ( Using normal approximation)

Therefore,

P( X> 20) = P( X > 20.5)

= P( Z > 20.5 - 12 / 2.6833)

= P( Z > 3.17)

= 0.0008