Question

p=0.3 20 students

exactly 2 get sick during the semester

Answer 1

X ~ Binomial (n,p)

Where n = 20 , p = 0.3

np = 20 * 0.3 = 6 > 5

nq = 20 * 0.7 = 14 > 5

Since np > 5 and nq > 5 , normal approximation can be used.

Using normal approximation with continuity correction,

P(X < x) = P( Z < x - np / sqrt(npq) )

P(X = 2) = P( 1.5 < X < 2.5)

= P( X < 2.5) - P( X < 1.5)

= P( Z < 2.5 - 0.3*20 / Sqrt( 20*0.3*0.7) ) - P( Z < 1.5 - 0.3*20 / sqrt(0.3*0.7*20) )

= P( Z < -1.7078) - P( Z < -2.1958)

= ( 1 - P( Z < 1.7078) ) - ( 1 - P( Z < 2.1958) )

= ( 1 - 0.9562) - ( 1 - 0.9859 )

= 0.0297