8 Out of 2000 voters 850 are republicans a Find exact probab

8) Out of 2000 voters 850 are republicans. a). Find exact probability that the sample of 100 contains at least 50 republicans. (no need to give in decimal form) b). Argue that you can approximate by Binomial probability and give approximation. c). Now argue that you can approximate by Normal and give approximation (in decimal form)

9) A stick is broken uniformly at random in 2 places. Find the probability that the pieces are the sides of a triangle.

10) A store is visited on average by 5 female and 7 male customers per hour. Find the probability that exactly 6 customers come in the next hour. (Hint: The type of the distribution remains the same).

Solution

8.

a)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative hypergeometric distribution table or technology, matching          
          
where          
N = population size =    2000      
K = number of successes in the population =    850      
n = sample size =    100      
x = critical number of successes in the sample =    50      
          
Thus,          
P(at most   49   ) =    0.926319228
Thus, the probability of at least   1   successes is  
          
P(at least   50   ) =    0.073680772 [ANSWER]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    100      
p = the probability of a success = 850/2000 =    0.425      
x = our critical value of successes =    50      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   49   ) =    0.921055521
          
Thus, the probability of at least   50   successes is  
          
P(at least   50   ) =    0.078944479 [ANSWER]

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c)

We first get the z score for the critical value:          
          
x = critical value =    49.5      
u = mean = np =    42.5      
          
s = standard deviation = sqrt(np(1-p)) =    4.943429983      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    1.416020865      
          
Thus, the left tailed area is          
          
P(z >   1.416020865   ) =    0.078384698 [ANSWER]

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