Question

According to a​ survey, 61​% of murders committed last year were cleared by arrest or exceptional means. Fifty murders committed last year are randomly​ selected, and the number cleared by arrest or exceptional means is recorded. When technology is​ used, use the Tech Help button for further assistance.

​(a) Find the probability that exactly 40 of the murders were cleared.

​(b) Find the probability that between 36 and 38 of the​ murders, inclusive, were cleared. ​

(c) Would it be unusual if fewer than 20 of the murders were​ cleared? Why or why​ not?

Answer 1

Condition check for Normal Approximation to Binomial
n * P >= 10 = 50 * 0.61 = 30.5
n * (1 - P ) >= 10 = 50 * ( 1 - 0.61 ) = 19.5

Using Normal Approximation to Binomial
Mean = n * P = ( 50 * 0.61 ) = 30.5
Variance = n * P * Q = ( 50 * 0.61 * 0.39 ) = 11.895
Standard deviation = √(variance) = √(11.895) = 3.4489

Part a)

P ( X = 40 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 40 - 0.5 < X < 40 + 0.5 ) = P ( 39.5 < X < 40.5 )

X ~ N ( µ = 30.5 , σ = 3.4489 )
P ( 39.5 < X < 40.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 39.5 - 30.5 ) / 3.4489
Z = 2.61
Z = ( 40.5 - 30.5 ) / 3.4489
Z = 2.9
P ( 2.61 < Z < 2.9 )
P ( 39.5 < X < 40.5 ) = P ( Z < 2.9 ) - P ( Z < 2.61 )
P ( 39.5 < X < 40.5 ) = 0.9981 - 0.9955
P ( 39.5 < X < 40.5 ) = 0.0027

Part b)

P ( 36 <= X <= 38 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 36 - 0.5 < X < 38 + 0.5 ) = P ( 35.5 < X < 38.5 )

X ~ N ( µ = 30.5 , σ = 3.4489 )
P ( 35.5 < X < 38.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 35.5 - 30.5 ) / 3.4489
Z = 1.45
Z = ( 38.5 - 30.5 ) / 3.4489
Z = 2.32
P ( 1.45 < Z < 2.32 )
P ( 35.5 < X < 38.5 ) = P ( Z < 2.32 ) - P ( Z < 1.45 )
P ( 35.5 < X < 38.5 ) = 0.9898 - 0.9265
P ( 35.5 < X < 38.5 ) = 0.0634

Part c)

P ( X < 20 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 20 - 0.5 ) = P ( X < 19.5 )

X ~ N ( µ = 30.5 , σ = 3.4489 )
P ( X < 19.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 19.5 - 30.5 ) / 3.4489
Z = -3.19
P ( ( X - µ ) / σ ) < ( 19.5 - 30.5 ) / 3.4489 )
P ( X < 19.5 ) = P ( Z < -3.19 )
P ( X < 19.5 ) = 0.0007

Yes, the event is unusual, since the probability is less than 0.05.