Question

A survey of U.S. adults found that 67% oppose raising the Medicare eligibility age from 65 to 67. You randomly select 80 U.S. adults S and ask them how they feel about raising the Medicare eligibility age from 65 to 67. Find the probability that the number who oppose raising the age is (a) at least 65, (b) exactly 50, and (c) more than 60, and (d) identify any unusual events. Explain.

Answer 1

0.477%

6.55%

5.05%

Event in question (a)

Explanation:

Given that:

p = 0.67 ; n = 80

Using Normal approximation :

Mean (m) = np = 80 * 0.67 = 53.6

Standard deviation (s) = sqrt(np(1-p)) = sqrt(80*0.67*0.33) = 4.2057

probability that the number who oppose raising the age is

(a) at least 65

65 - 0.5 = 64.5 = x

Z = (x - m) / s

Z = (64.5 - 53.6) / 4.2057 = 2.592

P(Z ≥ 2. 592) = 0.004771 (Z probability calculator)

(b) exactly 50

x = 50 + 0.5 = 50.5 ; x = 50 - 0.5 = 49.5

Z = (50.5 - 53.6) / 4.2057 = - 0.737

Z = (49.5 - 53.6) / 4.2057 = - 0.974

P(Z < - 0.737) - p(Z < 0.974) (Z probability calculator)

0.23056 - 0.16503 = 0.06553

(c) more than 60, and (d) identify any unusual events

x = 60 + 0.5 = 60.5

Z = (60.5 - 53.6) / 4.2057 = 1.64

P(Z > 1.64) = 0.0505

D.) Event with probability of less than 5%, which is event in (a) with probability of 0.477%