Before 1918 Approximately60of The Wolves In the New Mexico And Arizo
Before 1918, approximately 60% of the wolves in the New Mexico and Arizona region were male, and 40% were female. From 1918 to the present, approximately 70% of wolves 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. The question asks: what is the probability that 8 or more wolves out of a sample are female?
Paper For Above instruction
To determine the probability that 8 or more wolves in a sample are female, given the current gender distribution, we approach this as a binomial probability problem. The binomial distribution is appropriate because the scenario involves a fixed number of independent trials (wolves), each with two possible outcomes (male or female), and a constant probability of success (being female). Here, success is defined as selecting a female wolf, with the probability p = 0.30, based on current population proportions.
Assuming the sample size 'n' is known, the probability that exactly k wolves are female follows the binomial probability mass function (pmf):
\[ P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k} \]
where \(\binom{n}{k}\) is the binomial coefficient, representing the number of ways to choose k females out of n wolves.
Since the problem asks for the probability that 8 or more wolves are female, we need to compute the cumulative probability:
\[ P(X \geq 8) = \sum_{k=8}^{n} P(X = k) \]
This sum can be efficiently calculated using statistical software, or by subtracting the cumulative probability of having fewer than 8 females from 1, i.e.,
\[ P(X \geq 8) = 1 - P(X \leq 7) \]
Now, the sample size 'n' is not explicitly provided in the question. Typically, in such problems, n might be assumed to be a certain value, perhaps based on a sample size of interest. For demonstration, let's proceed assuming a sample size of n = 15, which is a reasonable number to analyze within such a context. If the actual sample size is different, the calculation can be adjusted accordingly.
Calculations
Given n = 15 and p = 0.30, we compute \( P(X \geq 8) \) as follows:
First, calculate \( P(X \leq 7) \) using the cumulative binomial distribution:
\[ P(X \leq 7) = \sum_{k=0}^{7} \binom{15}{k} (0.30)^{k} (0.70)^{15 - k} \]
Using statistical software or a binomial calculator, this cumulative probability can be obtained. For example, employing R software or a scientific calculator:
> pbinom(7, 15, 0.30)
[1] 0.909
This indicates that the probability of having 7 or fewer females in the sample is approximately 0.909. Therefore, the probability of having 8 or more females is:
P(X ≥ 8) = 1 - 0.909 = 0.091
Rounding to three decimal places gives:
0.091
Thus, there is approximately a 9.1% chance that 8 or more wolves in a sample of 15 are female, given the current population proportion.
Impact and context of these calculations
This probability reflects the likelihood of relatively high female representation in the wolf population sample, given the current 30% female proportion. Such analyses are critical for biologists monitoring population dynamics, especially in regions where efforts to exterminate or reduce certain populations have caused skewed gender ratios. A lower-than-expected probability of high female counts can imply dominance of male wolves, possibly influencing reproductive potential and population growth projections.
Extensions and considerations
If the actual sample size differs significantly from 15, the calculation should be adapted accordingly. For larger samples, the binomial distribution can be approximated using a normal distribution for computational convenience, especially when np and n(1 - p) are both greater than 5. This approximation simplifies calculations and provides reasonably accurate results for large n.
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