According to a survey, 63% of murders committed last year were cleared by arrest

Question

According to a survey, 63% 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. (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 20 of the murders were cleared? Why or why not? (a) The probability that exactly 38 of the murders were cleared is . (Round to four decimal places as needed.) (b) The probability that between 35 and 37 of the murders, inclusive, were cleared is . (Round to four decimal places as needed.) (c) Would it be unusual if fewer than 20 of the murders were cleared? Why or why not? A. No, it would not be unusual because 20 is between mu - 2 sigma and mu + 2 sigma. B. Yes, it would be unusual because 20 is between mu - 2 sigma and mu + 2 sigma. C. No, it would not be unusual because 20 is less than mu - 2 sigma. D. Yes, it would be unusual because 20 is less than mu - 2 sigma.

Explanation / Answer

n=50

P=0.63

Expectation = np = 31.5

Variance = np(1 - p) = 11.655

Standard deviation = 3.4139

a).

p( x=38)   = p( 37.5<x<38.5)

z value for 37.5, ( 37.5-31.5)/3.4139 =1.76

z value for 38.5, ( 38.5-31.5)/3.4139 =2.05

p( 1.76<z<2.05) = p( z<2.05)

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