Question

14.- A sociologist asserts that only 5% of all seniors in high school, capable of performing work at the university level, actually attend university. Find the probabilities that among 180 students capable of performing work at university level:
a) exactly 10 attend college using the binomial
b) Using the normal distribution
c) at least 10 go to university using binomial T.I or excel
d) Using the normal distribution
e) when many eight go to university using binomial or excel
f) Using the normal distribution

Answer 1

X ~ B ( n = 180 , P = 0.05 )

Using Normal Approximation to Binomial
Mean = n * P = ( 180 * 0.05 ) = 9
Variance = n * P * Q = ( 180 * 0.05 * 0.95 ) = 8.55
Standard deviation = √(variance) = √(8.55) = 2.924

Part a)

Part b)
P ( X = 10 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 10 - 0.5 < X < 10 + 0.5 ) = P ( 9.5 < X < 10.5 )

X ~ N ( µ = 9 , σ = 2.924 )
P ( 9.5 < X < 10.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 9.5 - 9 ) / 2.924
Z = 0.17
Z = ( 10.5 - 9 ) / 2.924
Z = 0.51
P ( 0.17 < Z < 0.51 )
P ( 9.5 < X < 10.5 ) = P ( Z < 0.51 ) - P ( Z < 0.17 )
P ( 9.5 < X < 10.5 ) = 0.695 - 0.5675
P ( 9.5 < X < 10.5 ) = 0.1275

Part c)

P ( X >= 10 ) = 1 - P ( X <= 9 ) = 1 - 0.5875 = 0.4125

Excel formula 1 - BINOM.DIST(9,180,0.05,TRUE)

Part d)
P ( X >= 10 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 10 - 0.5 ) =P ( X > 9.5 )

X ~ N ( µ = 9 , σ = 2.924 )
P ( X > 9.5 ) = 1 - P ( X < 9.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 9.5 - 9 ) / 2.924
Z = 0.17
P ( ( X - µ ) / σ ) > ( 9.5 - 9 ) / 2.924 )
P ( Z > 0.17 )
P ( X > 9.5 ) = 1 - P ( Z < 0.17 )
P ( X > 9.5 ) = 1 - 0.5675
P ( X > 9.5 ) = 0.4325

Excel formula 1-NORMSDIST(0.17)

Part e)

Excel formula  BINOM.DIST(8,180,0.05,FALSE)

Part f)
P ( X = 8 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 8 - 0.5 < X < 8 + 0.5 ) = P ( 7.5 < X < 8.5 )

X ~ N ( µ = 9 , σ = 2.924 )
P ( 7.5 < X < 8.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 7.5 - 9 ) / 2.924
Z = -0.51
Z = ( 8.5 - 9 ) / 2.924
Z = -0.17
P ( -0.51 < Z < -0.17 )
P ( 7.5 < X < 8.5 ) = P ( Z < -0.17 ) - P ( Z < -0.51 )
P ( 7.5 < X < 8.5 ) = 0.4325 - 0.305
P ( 7.5 < X < 8.5 ) = 0.1275

Excel formula  NORMSDIST(-0.51) - NORMSDIST(-0.17)