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 41 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 41 of the murders were cleared is. (b)The probability that between 35 and 37 of the murders, inclusive, were cleared is. (c)Would it be unusual if fewer than 20 of the murders were cleared? Why or why not? Yes, it would be unusual because 20 is between mu - 2sigma and mu + 2sigma. No, it would not be unusual because 20 is less than mu - 2sigma. No, it would not be unusual because 20 is between mu - 2sigma and mu. + 2sigma. Yes, it would be unusual because 20 is less than mu 2sigma.

Explanation / Answer

Here we will use binomial distribution p=0.63 and n=50

a. P(x=41)=ncxpx(1-p)n-x. I calculated using calculator=0.00193

b. P( 35<x<37)=P(x<37)-P(x<=35)=0.9310-0.8805=0.0505

c. Here mu=np=31.5 and sd=sqrt(np(1-p))=3.41

Now P(x<20)=0.0003

Now 20 is (31.5-20/3.41)=3.37 standard deviation below the mean

Answer is Yes,it would be unusual because 20 is less than mu-2sd

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