a)
X ~ bin ( n , p)
Where n = 35, p = 0.42
binomial probability distribution is
P(X) = nCx * px * ( 1 - p)n-x
P(X = 15 ) = 35C15 * 0.4215 * ( 1 - 0.42)20
= 0.1346
b)
Using Normal Approximation to Binomial
Mean = n * P = ( 35 * 0.42 ) = 14.7
Variance = n * P * Q = ( 35 * 0.42 * 0.58 ) = 8.526
Standard deviation = √(variance) = √(8.526) = 2.9199
P ( 14 < X < 16 ) = ?
Using continuity correction
P ( n + 0.5 < X < n - 0.5 ) = P ( 14 + 0.5 < X < 16 -
0.5 )
= P ( 14.5 < X < 15.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 14.5 - 14.7 ) / 2.9199
Z = -0.0685
Z = ( 15.5 - 14.7 ) / 2.9199
Z = 0.274
P ( -0.07 < Z < 0.27 )
P ( 14.5 < X < 15.5 ) = P ( Z < 0.27 ) - P ( Z < -0.07
)
P ( 14.5 < X < 15.5 ) = 0.608 - 0.4727
P ( 14.5 < X < 15.5 ) = 0.1353
c)
Rounded to 2 decimal places,
probability in part a = 0.13 , probability in part b = 0.14
both probabilities are not approximately same. (Since difference between probabilities are greater than 0.05)
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