According to a survey. 62% of murders committed last year were cleared by arrest

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. Find the probability that exactly 39 of the murders were cleared Find the probability that between 36 and 38 of the murders, inclusive, were cleared. Would it be unusual it fewer than 19 of the murders were cleared? Why or why not? The probability that exactly 39 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 19 of the murders were cleared? Why or why not? Yes, it would be unusual because 19 is between mu-2 sigma and mu + 2sigma. No, it would not be unusual because 19 is between mu - 2 alpha and mu + 2 sigma No. it would not be unusual because 19 is less than mu - 2 sigma. Yes, it would be unusual because 19 is less than mu - 2 sigma.

Explanation / Answer

Solution:

These are ALL Binomial probabilities with n = 50 and p = 0.62

(a) Find the probability that exactly 39 of the murders were cleared
P(39) = 50C39 (0.62)^39(1-0.62)^11= 3.73537388E+10 *1.90939356e-13
           = 0.00713229883

(b) Find the probability that between 36 and 38 of the murders, inclusive, were cleared.
  
   P(36) + P(37) + P(38)
  
   = 50C36 (0.62)^36(1-0.65)^14 + 50C37 (0.62)^37(1-0.62)^13 + 50C38            (0.62)^38(1-0.62)^12
   
   = 0.0130372146 + 0.0254528812 + 0.0142070791
   = 0.0526971749

(c) Would it be unusual if fewer than 19 of the murders were cleared?

   P(X < 19) = SUM x from 0 to 18: 50Cx (0.62)^x (1-0.62)^(50-x)
       =0.01261091746
  
   The mean = np = 50 * 0.62 = 31
   The standard deviation = sqrt (npq) = sqrt (50*0.62*0.38) = 11.78
    Now, 19 is (31 - 19) / 11.78 = 1.01 standard deviations BELOW the mean
  
   Answer: D.Yes, it would be unusual because 19 is less than u-2

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