Question

27% of all college students major in STEM (Science, Technology, Engineering, and Math). If 45 college students are randomly selected, find the probability that

a. Exactly 13 of them major in STEM.

b. At most 12 of them major in STEM.

c. At least 9 of them major in STEM.

d. Between 8 and 15 (including 8 and 15) of them major in STEM.

Answer 1

Using Normal Approximation

Mean = n * P = ( 45 * 0.27 ) = 12.15
Variance = n * P * Q = ( 45 * 0.27 * 0.73 ) = 8.8695
Standard deviation = √(variance) = √(8.8695) = 2.9782

Part a)

P ( X = 13 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 13 - 0.5 < X < 13 + 0.5 ) = P ( 12.5 < X < 13.5 )

X ~ N ( µ = 12.15 , σ = 2.9782 )
P ( 12.5 < X < 13.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 12.5 - 12.15 ) / 2.9782
Z = 0.12
Z = ( 13.5 - 12.15 ) / 2.9782
Z = 0.45
P ( 0.12 < Z < 0.45 )
P ( 12.5 < X < 13.5 ) = P ( Z < 0.45 ) - P ( Z < 0.12 )
P ( 12.5 < X < 13.5 ) = 0.6736 - 0.5478
P ( 12.5 < X < 13.5 ) = 0.1259

Part b)

P ( X <= 12 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 12 + 0.5 ) = P ( X < 12.5 )

X ~ N ( µ = 12.15 , σ = 2.9782 )
P ( X < 12.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 12.5 - 12.15 ) / 2.9782
Z = 0.12
P ( ( X - µ ) / σ ) < ( 12.5 - 12.15 ) / 2.9782 )
P ( X < 12.5 ) = P ( Z < 0.12 )
P ( X < 12.5 ) = 0.5478

Part c)

P ( X >= 9 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 9 - 0.5 ) =P ( X > 8.5 )

X ~ N ( µ = 12.15 , σ = 2.9782 )
P ( X > 8.5 ) = 1 - P ( X < 8.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 8.5 - 12.15 ) / 2.9782
Z = -1.23
P ( ( X - µ ) / σ ) > ( 8.5 - 12.15 ) / 2.9782 )
P ( Z > -1.23 )
P ( X > 8.5 ) = 1 - P ( Z < -1.23 )
P ( X > 8.5 ) = 1 - 0.1093
P ( X > 8.5 ) = 0.8907

Part d)

P ( 8 <= X <= 15 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 8 - 0.5 < X < 15 + 0.5 ) = P ( 7.5 < X < 15.5 )

X ~ N ( µ = 12.15 , σ = 2.9782 )
P ( 7.5 < X < 15.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 7.5 - 12.15 ) / 2.9782
Z = -1.56
Z = ( 15.5 - 12.15 ) / 2.9782
Z = 1.12
P ( -1.56 < Z < 1.12 )
P ( 7.5 < X < 15.5 ) = P ( Z < 1.12 ) - P ( Z < -1.56 )
P ( 7.5 < X < 15.5 ) = 0.8686 - 0.0594
P ( 7.5 < X < 15.5 ) = 0.8093