find the probability

Question

34% of all college students major in STEM (Science, Technology, Engineering, and Math). If 36 college students are randomly selected, find the probability that

a. Exactly 13 of them major in STEM. ?

b. At most 13 of them major in STEM. ?

c. At least 10 of them major in STEM. ?

d. Between 12 and 20 (including 12 and 20) of them major in STEM. ?

Answer 1

use binomial approximation:

mean = np = 36 * 0.34 = 12.24

standard deviation = sqrt(npq) = sqrt(36 * 0.34 * 0.66) = 2.84

z = (x - mean)/SD

z = (x - 12.24)/2.84

a) P(x = 13) = P(12.5 < x < 13.5) = P((12.5-12.24)/2.84 < z < (13.5- 12.24)/2.84) = P(0.197 < z < 0.443) = 0.0907.

b) P(x <= 13) = P(x < 13.5) = P( z < (13.5 - 12.24)/2.84) = P(z < 0.4437) = 0.57

c) P(x >= 10) = P(x >= 9.5) = P( z> (9.5 - 12.24)/2.84) = P(z > -0.9648) = 0.8315

d) P(12 <= x <= 20) = P(11.5 <=x <= 20.5) = P( (11.5-12.24)/2.84 < z < (20.5-12.24/2.84)) = P(-2.61 < z < 2.91) = 0.9937