Question Part Points Submissions Used Before 1918, approximately 55% of the wolv

Question

Question Part Points Submissions Used Before 1918, approximately 55% of the wolves in the New Mexico and Arizona region were male, and 45% were female. However, cattle ranchers in this area have made a determined effort to exterminate wolves. From 1918 to the present, approximately 65% of wolves in the region are male, and 35% are female. Biologists suspect that male wolves are more likely than females to return to an area where the population has been greatly reduced. (Round your answers to three decimal places.)

(a) Before 1918, in a random sample of 9 wolves spotted in the region, what is the probability that 6 or more were male?


What is the probability that 6 or more were female?
  

What is the probability that fewer than 3 were female?
  

(b) For the period from 1918 to the present, in a random sample of 9 wolves spotted in the region, what is the probability that 6 or more were male?


What is the probability that 6 or more were female?


What is the probability that fewer than 3 were female?

Explanation / Answer

a)

Before 1918, in a random sample of 9 wolves spotted in the region, what is the probability that 6 or more were male?

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 =    9      
p = the probability of a success =    0.55      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.638615396
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.361384604 [ANSWER]

***************************

What is the probability that 6 or more were female?

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 =    9      
p = the probability of a success =    0.45      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.834177952
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.165822048 [ANSWER]

*****************************

What is the probability that fewer than 3 were female?

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    9      
p = the probability of a success =    0.45      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.14950314
          
Which is also          
          
P(fewer than   3   ) =    0.14950314 [ANSWER]

**************************************************************************************

(b) For the period from 1918 to the present, in a random sample of 9 wolves spotted in the region, what is the probability that 6 or more were male?

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 =    9      
p = the probability of a success =    0.65      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.391105587
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.608894413 [ANSWER]

***************************

What is the probability that 6 or more were female?

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 =    9      
p = the probability of a success =    0.35      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.9464118
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.0535882 [ANSWER]

***************************

What is the probability that fewer than 3 were female?

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    9      
p = the probability of a success =    0.35      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.337273279
          
Which is also          
          
P(fewer than   3   ) =    0.337273279 [ANSWER]

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