Question

Courts sometimes make mistakes, but which do you believe is the worse mistake: convicting an innocent person or letting a guilty person go free? It turns out that about 60% of all Americans believe that convicting an innocent person is the worse mistake.† Suppose you are taking a sociology class with 27 students enrolled. The question discussed today is: Do you agree with the statement that convicting an innocent person is worse than letting the guilty go free? (Round your answers to four decimal places.)

(a) What is the probability that the proportion of the class who agree is at least one half?


(b) What is the probability that the proportion of the class who agree is at least two thirds?


(c) What is the probability that the proportion of the class who agree is no more than one third?


(d) Is the normal approximation to the proportion p̂ = r/n valid? Explain.

---Select--- Yes No ,  ---Select--- nq exceeds both np and nq exceed np exceeds np and nq do not exceed np does not exceed nq does not exceed

Answer 1

Answer:

Given,

sample n = 27

p = 0.60

Mean = p

= 0.6

Standard deviation = sqrt(pq/n)

= sqrt(0.6*0.4/27)

= 0.0943

a)

P(p > 0.5) = P(z > (0.5 - 0.6)/0.0943)

= P(z > -1.06)

= 0.8554277 [since from z table]

= 0.8554

b)

P(p > 0.67) = P(z > (0.67 - 0.6)/0.0943)

= P(z > 0.7423)

= 0.2289528 [since from z table]

= 0.2290

c)

P(p < 0.33) = P(z < (0.33 - 0.6)/0.0943)

= P(z < -2.8632)

= 0.0020969 [since from z table]

= 0.0021

d)

np = 27*0.6

= 16.2

nq = 27*0.4

= 10.8

Yes, the normal approximation is valid, np & nq > 10