Question

3) A tutoring company claims that 75% of the high school students who hire one of 3)- their tutors will improve their grades. a. In a sample of 100 high school students, what is the probability that 80% or more improved their grades? b. In a sample of 200 high school students, what is the probability that 80% or more improved their grades?

Answer 1

Using central limit theorem,

P($\hat{p}$ < p) = P( Z < $\hat{p}$ - p / sqrt( p( 1 -p ) / n)

a)

P($\hat{p}$ >= 0.80) = P( Z >= 0.80 - 0.75 / sqrt( 0.75 * 0.25 / 100) )

= P( Z >= 1.1547)

= 0.1241

b)

P($\hat{p}$ >= 0.80) = P( Z >= 0.80 - 0.75 / sqrt( 0.75 * 0.25 / 200) )

= P( Z >= 1.6330)

= 0.0512