Question

According to a​ survey, 62​% 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 38 of the murders were cleared.​

(b) Find the probability that between 35 and 37 of the​ murders, inclusive, were cleared.

​(c) Would it be unusual if fewer than 19 of the murders were​ cleared? Why or why​ not?

Answer 1

X ~ Bin( n , p)

Where n = 50 , = 0.62

Mean = np = 50 * 0.62 = 31

Standard deviation = sqrt (np(1-p) )

= sqrt( 50 * 0.62 ( 1 - 0.62) )

= 3.4322

Using normal approximation

P(X < x) = P(Z < x - Mean / SD)

a)

P(X = 38) = P(37.5 < X < 38.5) (With continuity correction )

= P(X < 38.5) - P(X < 37.5)

= P(Z < (38.5 - 31) / 3.4322) - P(Z < (37.5 - 31) / 3.4322)

= P(Z < 2.19) - P(Z < 1.89)

= 0.9857 - 0.9706

= 0.0151

b)

P(35 < X < 37) = P(34.5 < X < 37.5)

=P(X < 37.5) - P(X < 35.5)

= P(Z < (37.5 - 31) / 3.4322) - P(Z < (34.5 - 31) / 3.4322)

= P(Z < 1.89) - P(Z < 1.02)

= 0.9706 - 0.8461

= 0.1245

c)

P(X < 19) = P(Z < (19 - 31) / 3.4322)

= P(Z < -3.50)

= 0.0002

Since this probability is less than 0.05 , the event is unusual.