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68.7% of the population has brown eyes. A convention has 6000 people in attendance. Find the standard deviation for the number of brown-eyed people at such gatherings of 6000 people.

Select one:
a. 4122.0
b. 35.9
c. 1296.0
d. 36.0

Answer :

The standard deviation for the number of brown-eyed people at a convention of 6000 people, when 68.7% of them have brown eyes, is calculated to be approximately 35.9, corresponding to option (b).

To find the standard deviation for the number of brown-eyed people at a convention of 6000 people when 68.7% of the population have brown eyes, we need to first find the expected number of brown-eyed individuals, which we can calculate by multiplying 6000 by 0.687. Then, we use the formula for the standard deviation of a binomial distribution, which is \\( \sigma = \sqrt{npq} \)\, where \(n\) is the total number of trials (people at the convention), \(p\) is the probability of success (having brown eyes), and \(q\) is the probability of failure (not having brown eyes).

First, find \(q\) which is \(1 - p\) or \(1 - 0.687 = 0.313\). Then we plug the numbers into the formula:

\( \sigma = \sqrt{6000 \times 0.687 \times 0.313} \)

Calculating this gives us:

\( \sigma \approx \sqrt{6000 \times 0.687 \times 0.313} \approx \sqrt{6000 \times 0.2151} \approx \sqrt{1290.6} \approx 35.93 \)

The standard deviation for the number of brown-eyed people at the convention is therefore around 35.9, which corresponds to option (b).

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