In the following problem check that it is appropriate to use

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. What are the chances that a person who is murdered actually knew the murderer? The answer to this question explains why a lot of police detective work begins with relatives and friends of the victim! About 67% of people who are murdered actually knew the person who committed the murder. Suppose that a detective file in New Orleans has 65 current unsolved murders. Find the following probabilities. (Round your answers to four decimal places.)

(a) at least 35 of the victims knew their murderers

(b) at most 48 of the victims knew their murderers

(c) fewer than 30 victims did not know their murderers

(d) more than 20 victims did not know their murderers

Solution

a)

We first get the z score for the critical value:          
          
x = critical value =    34.5      
u = mean = np =    43.55      
          
s = standard deviation = sqrt(np(1-p)) =    3.790976128      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -2.387247953      
          
Thus, the left tailed area is          
          
P(z >   -2.387247953   ) =    0.991512481 [ANSWER]

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b)

We first get the z score for the critical value:          
          
x = critical value =    48.5      
u = mean = np =    43.55      
          
s = standard deviation = sqrt(np(1-p)) =    3.790976128      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    1.305732306      
          
Thus, the left tailed area is          
          
P(z <   1.305732306   ) =    0.904178196 [ANSWER]

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c)

The probability of the victim not knowing the murderer is 1 -0.67 = 0.33.

We first get the z score for the critical value:          
          
x = critical value =    29.5      
u = mean = np =    21.45      
          
s = standard deviation = sqrt(np(1-p)) =    3.790976128      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    2.123463649      
          
Thus, the left tailed area is          
          
P(z <   2.123463649   ) =    0.983142491 [ANSWER]

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d)

We first get the z score for the critical value:          
          
x = critical value =    20.5      
u = mean = np =    21.45      
          
s = standard deviation = sqrt(np(1-p)) =    3.790976128      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -0.250595089      
          
Thus, the left tailed area is          
          
P(z <   -0.250595089   ) =    0.59893641 [ANSWER]