According to a survey, 64% of murders committed last year were cleared by arrest

Question

According to a survey, 64% 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. Find the probability that exactly 41 of the murders were cleared. Find the probability that between 36 and 38 of the murders, inclusive, were cleared. Would it be unusual if fewer than 18 of the murders were cleared? Why or why not? The probability that exactly 41 of the murders were cleared is. (Round to four decimal places as needed.) The probability that between 36 and 38 of the murders, inclusive, were cleared is. (Round to four decimal places as needed.) Would it be unusual if fewer than 18 of the murders were cleared? Why or why not? No, it would not be unusual because 18 is less than mu - 2sigma. Yes, it would be unusual because 18 is less than 2sigma. No, it would not be unusual because 18 is between mu - 2sigma and mu + 2sigma. Yes, it would be unusual because 18 is between mu - 2sigma and mu + 2sigma.

Explanation / Answer

This corresponds to binomial probability with probability of success, p=0.64, number of trials, n=50 and r=41 being the specific number of success in 50 trials. Use, P(X,r)=nCr(p)^r(q)^n-r

a) P(X=41)=50C41(0.64)^41(0.36)^9=0.0029

b) P(36<X<38)

Compute Z score corresponding to 36 and 38.

mean=np=50*0.64=32, sd=sqrt npq=sqrt 50*0.64*0.36=3.39

z1=(36-32)/3.39=1.18 and z2=1.77

P(36<X<38)=0.3810+0.4616=0.8426

c) A

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