Before 1918, approximately 55% of the wolves in the New Mexico and Arizona regio

Question

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 12 wolves spotted in the region, what is the probability that 9 or more were male?

What is the probability that 9 or more were female?

What is the probability that fewer than 6 were female?

(b) For the period from 1918 to the present, in a random sample of 12 wolves spotted in the region, what is the probability that 9 or more were male?

What is the probability that 9 or more were female?

What is the probability that fewer than 6 were female?

Explanation / Answer

Before 1918, the proportion of male wolve = p = 0.55

n = sample size = 12

we want to find the probability that 9 or more were male out of 12.

By using binomial distribution formula:

Let X be the number of male wolve

Then we want to find P( X >= 9) = 1 - P( X < 9) = 1 - P(X <= 8) ....(1)

Let's use excel to find less than binomial probability.

P( X < = 8) = "=BINOMDIST(8,12,0.55,1)" = 0.8655

Plug this in equation 1), we get

P(X > = 9) = 1 - 0.8655 = 0.1345

Before 1918, the proportion of female wolve = p = 0.45

we want to find the probability that 9 or more were female out of 12.

Let Y be the number of male wolve

Then we want to find P( Y >= 9) = 1 - P( Y < 9) = 1 - P(Y <= 8) ....(2)

P( Y < = 8) = "=BINOMDIST(8,12,0.45,1)" = 0.9644

Plug this in equation 2), we get

P(X > = 9) = 1 - 0.8655 = 0.0356

Next we want to find the probability that fewer than 6 were female

P( Y < 6) = P(Y <= 5) = "=BINOMDIST(5,12,0.45,1)" = 0.5269

After 1918, the proportion of male wolve = p = 0.65

n = sample size = 12

we want to find the probability that 9 or more were male out of 12.

Let X be the number of male wolve

Then we want to find P( X >= 9) = 1 - P( X < 9) = 1 - P(X <= 8) ....(3)

P( X < = 8) = "=BINOMDIST(8,12,0.65,1)" = 0.6533

Plug this in equation 3), we get

P(X > = 9) = 1 - 0.8655 = 0.3467

After 1918, the proportion of female wolve = p = 0.35

we want to find the probability that 9 or more were female out of 12.

Let Y be the number of male wolve

Then we want to find P( Y >= 9) = 1 - P( Y < 9) = 1 - P(Y <= 8) ....(4)

P( Y < = 8) = "=BINOMDIST(8,12,0.35,1)" = 0.9944

Plug this in equation 4), we get

P(X > = 9) = 1 - 0.8655 = 0.0056

Next we want to find the probability that fewer than 6 were female

P( Y < 6) = P(Y <= 5) = "=BINOMDIST(5,12,0.35,1)" = 0.7873

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