Question

Ella Educationer is Edith’s sister. She is also interested in studying children’s programming. Ella watches programs and then rates them as either Good or Bad. She believes that this now becomes a binomial experiment, with a constant probability across every new program viewed. Suppose Ella is going to watch 10 programs and we find the probability that a show will be rated as good is 0.40. Ella next sampled 100 programs and is interested in calculating the probability that at least half of the shows were rated good P(X ≥ 50). Use the normal approximation to the binomial to calculate this probability. Note: p = .40

Please show your work.

Answer 1

Given that

p = 0.40

n = 100

using normal to binomial approximation, we get

Mean= n*p

= 100*0.40

= 40

and

standard deviation = sqrt(n*p*(1-p))

= sqrt(100*0.40*0.60)

= 4.90

We have to find [using continuity correction, we write P(at least 50) = P(X>50-0.5)]

P(X>x) = P(z>(x-mean)/sd)

= P(z>(49.5-40)/4.90)

= P(z>(9.5/4.90))

=P(z>1.94)

= 1 - P(z<1.94) ......................[using P(z>a) =1 -P(z<a)]

= 1 - 0.9738.....[using z table, check 1.9 in the left most column and 0.04 in the top row, then select intersecting cell]

= 0.0262