A researcher is studying the average wing length of a particular species of butterfly, known to have a Normally distributed wing length in the population.

The researcher randomly selects a sample of n = 8 butterflies from the population.
They calculate point estimates for the population mean wing length and population standard deviation based on the sample.
Using these estimates, they compute the standard error of the sample mean wing length.
Assuming this sampling procedure is repeated in future studies, what is the probability that the sample mean wing length will fall within 1.89 estimated standard errors of the true population mean wing length?

Question

A researcher is studying the average wing length of a particular species of butterfly, known to have a Normally distributed wing length in the population.

The researcher randomly selects a sample of n = 8 butterflies from the population.
They calculate point estimates for the population mean wing length and population standard deviation based on the sample.
Using these estimates, they compute the standard error of the sample mean wing length.
Assuming this sampling procedure is repeated in future studies, what is the probability that the sample mean wing length will fall within  1.89 estimated standard errors of the true population mean wing length?

BlackTom AI · Accepted Answer

The question describes a butterfly wing length that is normally distributed in the population and a sample of size n = 8. We are asked for the probability that the sample mean will fall within 1.89 estimated standard errors of the population mean.

First, identify the appropriate sampling distribution. When using the sample standard deviation s to estimate the standard error of the sample mean, the statistic (X̄ − μ) / (s / √n) follows a t distribution with df = n − 1. Here, n − 1 = 7, so we should use a t7 distribution.

Define the event of interest: |X̄ − μ| ≤ 1.89 · (s / √n). This is equivalent to |T| ≤ 1.89 where T ~ t7.

Next, translate this into a probability using the t distribution. The probability P(|T| ≤ 1.89) equals F_t7(1.89) − F_t7(−1.89). By symmetry, this is 2 · F_t7(1.89) − 1.

Consulting the t7 distribution (or recalling typical critical values): the two-sided 0.10 central region corresponds to critical values around ±1.895, since the two-sided alpha = 0.10 uses t7 ≈ 1.895. This means F_t7(1.89) is very close to 0.95. Therefore, the probability is approximately 2 × 0.95 − 1 ≈ 0.90.

In other words, about 90% of repeated samples of size 8 would have their sample mean fall within 1.89 estimated standard errors of the true mean, reflecting the spread described by the t7 distribution.

Note: If one instead used the true known standard error (i.e., replacing s with the population standard deviation σ), the calculation would use a normal reference rather than a t distribution, but the problem explicitly uses the estimated standard error, which leads to the t7-based conclusion above.

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