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Ecology: Wolves The following is based on information taken from The Wolf in the Southwest: The Making of an Endangered Species, edited by David Brown (University of Arizona Press). 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 \(70 \%\) of wolves in the region are male, and \(30 \%\) are female. Biologists suspect that male wolves are more likely than females to return to an area where the population has been greatly reduced. (a) Before 1918 , in a random sample of 12 wolves spotted in the region, what was the probability that 6 or more were male? What was the probability that 6 or more were female? What was the probability that fewer than 4 were female? (b) Answer part (a) for the period from 1918 to the present.

Step by step solution

01

Define the problem

We need to determine the probabilities for a specific number of male or female wolves out of a sample, before and after 1918. This is a binomial probability problem. Before 1918, the probabilities are 55% male (p = 0.55) and 45% female. After 1918, they are 70% male (p = 0.70) and 30% female.

02

Set up binomial distribution for males before 1918

The number of males in a sample of 12 wolves is a binomial random variable given by \( B(n=12, p=0.55) \). We want to find the probability that 6 or more wolves are male: \( P(X \geq 6) = 1 - P(X < 6) = 1 - [P(X = 0) + P(X = 1) + \ldots + P(X = 5)] \).

03

Calculate binomial probability for males before 1918

Using the binomial probability formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), calculate \( P(X = 0) \) to \( P(X = 5) \) and subtract the sum from 1 to find \( P(X \geq 6) \). The calculations involve using the formula for each k value and obtaining the result by summing the probabilities for the cases from 0 to 5.

04

Set up binomial distribution for females before 1918

The number of females is given by \( B(n=12, p=0.45) \). To find the probability that 6 or more are female, calculate \( P(X \geq 6) \) similarly: subtract the sum of probabilities from 0 to 5 from 1.

05

Calculate binomial probability for females less than 4 before 1918

For fewer than 4 females, calculate \( P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \) using the binomial formula and add these probabilities.

06

Repeat the calculations for males after 1918

With \( p = 0.70 \) for males after 1918, repeat steps 2-3: set up the distribution \( B(n=12, p=0.70) \) and find \( P(X \geq 6) \). Use the formula and process previously used.

07

Repeat the calculations for females after 1918

For females after 1918 with \( p = 0.30 \), repeat steps 4 and 5 to find 1) the probability of 6 or more females and 2) the probability of fewer than 4 females. Use the results for \( B(n=12, p=0.30) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution

In the world of statistics, a binomial distribution is a common setup when you need to figure out probabilities in situations that are either "yes or no," like flipping a coin or, in this case, spotting male versus female wolves. A binomial distribution is applicable when there are:

• A fixed number of trials, which in our exercise would be spotting a fixed number of wolves, say 12.
• Each trial has two outcomes - male or female wolf.
• The probability of each outcome remains constant across every trial. Before 1918, the chance a wolf was male was 55%, whereas, post-1918, it increased to 70%.
• Each wolf spotted is independent of others in terms of being male or female.

These foundational points allow us to use binomial distribution modeling to calculate probabilities of spotting a certain number of male or female wolves in a given sample.

Probability Calculation

Probability calculations involve using mathematical models to predict how likely it is that a particular event will occur. When dealing with a binomial distribution, there's a specific formula that gets the job done here. The binomial probability formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] This formula helps in calculating the probability of exactly "k" successes in "n" trials, where "p" is the probability of success in a single trial. For example, figuring out the probability of spotting 6 or more male wolves before 1918 involves:

• Calculating individual probabilities for having 0 to 5 male wolves and subtracting their sum from 1 to find the probability for 6 or more.
• Using constants like \( n = 12 \) and \( p = 0.55 \) (or \( p = 0.70 \) post-1918), based on the scenario.

Breaking it down with this formula helps us systematically determine the likelihood of either males or females outnumbering the other based on historical data.

Statistical Analysis

Statistical analysis is a crucial process that involves collecting, analyzing, interpreting, presenting, and organizing data. In the context of our wolf scenario, statistics can highlight changes in population dynamics over time.

• By calculating probabilities before and after 1918, we can infer effects of external factors, like rancher influence, on wolf populations.
• Statistical insights can also support theories, such as biologists' suspicions that male wolves might be more resilient in certain environments.

The statistical analysis takes the raw probabilities we've calculated and uses them to frame broader narratives or conclusions, such as shifts in gender ratios over time due to human activities. This not only provides a mathematical view but also serves as a lens into historical and environmental impacts on wildlife.