Question

52. According to the High School Athletics Participation Survey, approximately 55% of students enrolled in high schools participate in athletic programs. You are preforming a study of high school students and would like at least 11 students in the study participating in athletics.

a. How many high school students do you expect to have to randomly select?

b. How many high school students do you have to select to have a 99% probability that the sample contains at least 12 who participate in athletics?

Answer 1

a) Randomly we should select at least (100 * 11)/55 = 20 students.

b)

suppose sample size is n

P(X >= 12) >= 0.99

p= 0.55

n	P(X>=12)
12	0.000766
13	0.004904
14	0.017006
15	0.042421
16	0.085309
17	0.147068
18	0.22581
19	0.316926
20	0.414306
21	0.511686
22	0.603711
23	0.686533
24	0.757967
25	0.817312
26	0.865
27	0.902196
28	0.930442
29	0.951378
30	0.966556
31	0.97734
32	0.984863
33	0.990021
34	0.993502

Excel formulas

n	P(X>=12)
12	=1-BINOM.DIST(11,A2,0.55,1)
13	=1-BINOM.DIST(11,A3,0.55,1)
=1+A3	=1-BINOM.DIST(11,A4,0.55,1)
=1+A4	=1-BINOM.DIST(11,A5,0.55,1)
=1+A5	=1-BINOM.DIST(11,A6,0.55,1)
=1+A6	=1-BINOM.DIST(11,A7,0.55,1)
=1+A7	=1-BINOM.DIST(11,A8,0.55,1)
=1+A8	=1-BINOM.DIST(11,A9,0.55,1)
=1+A9	=1-BINOM.DIST(11,A10,0.55,1)
=1+A10	=1-BINOM.DIST(11,A11,0.55,1)
=1+A11	=1-BINOM.DIST(11,A12,0.55,1)
=1+A12	=1-BINOM.DIST(11,A13,0.55,1)
=1+A13	=1-BINOM.DIST(11,A14,0.55,1)
=1+A14	=1-BINOM.DIST(11,A15,0.55,1)
=1+A15	=1-BINOM.DIST(11,A16,0.55,1)
=1+A16	=1-BINOM.DIST(11,A17,0.55,1)
=1+A17	=1-BINOM.DIST(11,A18,0.55,1)
29	=1-BINOM.DIST(11,A19,0.55,1)
30	=1-BINOM.DIST(11,A20,0.55,1)
31	=1-BINOM.DIST(11,A21,0.55,1)
32	=1-BINOM.DIST(11,A22,0.55,1)
33	=1-BINOM.DIST(11,A23,0.55,1)
34	=1-BINOM.DIST(11,A24,0.55,1)

min n = 33