Question

A multiple-choice test consists of 25 questions, each with four possible answers. If our desperate business statistics student answers the test by guessing the answers before reading the questions (that is, by selecting an answer at random for each question), use the normal curve approximation to find the probabilities that he will get:

exactly six correct answers (b) at least seven correct answers

fewer than four correct answers (d) more than two, but fewer than nine correct answers

What is the probability of getting at least 12 replies to questionnaires mailed to 100 persons, when the probability is 0.18 that any one of them will reply?

Answer 1

n = 25

P = 0.25

= n * P = 25 * 0.25 = 6.25

= sqrt(n * P * (1 - P))

= sqrt (25 * 0.25 * 0.75) = 4.6875

a) P(X = 6)

= P(5.5 < X < 6.5)

= P((5.5 - )/< (X - )/< (6.5 - )/)

= P((5.5 - 6.25)/4.6875 < Z < (6.5 - 6.25)/4.6875)

= P(-0.16 < Z < 0.05)

= P(Z < 0.05) - P(Z < -0.16)

= 0.5199 - 0 .4364

= 0.0835

B) P(X > 7)

= P(X > 6.5)

= P((X - )/> (6.5 - )/)

= P(Z > (6.5 - 6.25)/4.6875)

= P(Z > 0.05)

= 1 - P(Z < 0.05)

= 1 - 0.5199

= 0.4801

C) P(X < 4)

= P(X < 3)

= P(X < 3.5)

= P((X - )/< (3.5 - )/)

= P(Z < (3.5 - 6.25)/4.6875)

= P(Z < -0.59)

= 0.2776

D) P(2 < X < 9)

= P(3 < X < 8)

= P(2.5 < X < 8.5)

= P((2.5 - )/< (X - )/< (8.5 - )/)

=P((2.5 - 6.25)/4.6875 < Z < (8.5 - 6.25)/4.6875)

= P(-0.8 < Z < 0.48)

= P(Z < 0.48) - P(Z < -0.8)

= 0.6844 - 0.2119

= 0.4725