Question

33% of all college students major in STEM (Science, Technology, Engineering, and Math). If 34 college students are randomly selected, find the probability that a. Exactly 12 of them major in STEM. b. At most 10 of them major in STEM. c. At least 9 of them major in STEM. d. Between 7 and 15 (including 7 and 15) of them major in STEM.

Answer 1

33% of all college students major in STEM (Science, Technology, Engineering, and Math). If 34 college students are randomly selected, find the probability that

This is binomial distribution with n=34, p=0.33

Since np=11.22 and n(1-P)=22.78 both are > 10, normal approximation to binomial can be used.

Expectation = np = 11.22

Variance = np(1 - p) = 7.5174

Standard deviation = 2.7418

We are using with continuity correction for calculating all z values.

a. Exactly 12 of them major in STEM

z value for 11.5, z =(11.5-11.22)/2.7418 =0.10

z value for 12.5, z =(12.5-11.22)/2.7418 =0.47

P( x=12)= P( 11.5<x<12.5) = P(0.10<z<0.47)

=P( z < 0.47)-P( z <0.10)

=0.6808 -0.5398

=0.141

b. At most 10 of them major in STEM.

. At most 10 is ≤10.

Z value of 10.5, z =(10.5-11.22)/2.7418 =-0.26

P(x ≤ 10) =p( z < -0.26)

=0.3974

c. At least 9 of them major in STEM.

. At least 9 is ≥9

Z value of 8.5, z =(8.5-11.22)/2.7418 =-0.99

P(x ≥ 9 ) =p( z > -0.99)

=0.8389

d. Between 7 and 15 (including 7 and 15) of them major in STEM.

z =(6.5-11.22)/2.7418 =-1.72

z =(15.5-11.22)/2.7418 =1.56

P( 7≤ x≤ 15) = P( -1.72<z<1.56)

P( z < 1.56)-P( z < -1.72)

=0.9406-0.0427

=0.8979

Excel function used for calculating p values. For example p( z < -0.26)

Excel function: =NORM.S.DIST(-0.26,TRUE)

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