In a large population of mesquite trees, 13% are infested with mistletoe. A biologist selects a random sample of 25 mesquite trees from this population. About how far do you expect the sample proportion of infested trees to vary from the true proportion, on average?

(a) 0.005
(b) 0.026
(c) 0.067
(d) 0.072
(e) 0.130

Question

06-06-2025
Mathematics

High School

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In a large population of mesquite trees, 13% are infested with mistletoe. A biologist selects a random sample of 25 mesquite trees from this population. About how far do you expect the sample proportion of infested trees to vary from the true proportion, on average?

(a) 0.005
(b) 0.026
(c) 0.067
(d) 0.072
(e) 0.130

Answer :

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MahershalaAli MahershalaAli · Answer 1 · 2025-02-13T14:42:25-05:00

Final answer:

The expected variability of the sample proportion from the true proportion of mesquite trees infested with mistletoe, given a sample size of 25, is calculated using the standard error formula. So,option the correct answer is c) 0.067.

Explanation:

The question is asking about the expected variability, or standard error, of the sample proportion from the true population proportion of mesquite trees infested with mistletoe. The true proportion (p) is 13% or 0.13, and the sample size (n) is 25 trees.

To calculate the standard error of the sample proportion, we use the formula: \(SE = \sqrt{\frac{p(1-p)}{n}}\). Plugging in the values, we get:

\(SE = \sqrt{\frac{0.13(1-0.13)}{25}} = \sqrt{\frac{0.13(0.87)}{25}} = \sqrt{\frac{0.1131}{25}} = \sqrt{0.004524} = 0.067\).

Thus, we expect the sample proportion of infested trees to vary from the true proportion by an average of 0.067, making option (c) the correct answer.