A scientist is investigating the average rate of enzyme activity in a species of freshwater algae under controlled lab conditions. The rate of activity is known to follow a Normal distribution in the population.

The scientist randomly selects a sample of n = 9 algae specimens.
They calculate point estimates for the mean enzyme activity rate and the standard deviation of the population based on the sample.
They then estimate the standard error of the sample mean enzyme activity rate.
Assuming this sampling method is used in future experiments, what is the probability that the sample mean enzyme activity rate will fall within .6 estimated standard errors of the true population mean?

Question

A scientist is investigating the average rate of enzyme activity in a species of freshwater algae under controlled lab conditions. The rate of activity is known to follow a Normal distribution in the population.

The scientist randomly selects a sample of n = 9 algae specimens.
They calculate point estimates for the mean enzyme activity rate and the standard deviation of the population based on the sample.
They then estimate the standard error of the sample mean enzyme activity rate.
Assuming this sampling method is used in future experiments, what is the probability that the sample mean enzyme activity rate will fall within .6 estimated standard errors of the true population mean?

BlackTom AI · Accepted Answer

We begin by restating what is being asked in plain terms: we want the probability that the sample mean from n = 9 algae specimens will lie within 0.6 estimated standard errors of the true population mean, given that the population rate of enzyme activity is normally distributed and that the standard error is estimated from the sample.
First, recall how the sampling distribution behaves when the population standard deviation is unknown. If the population is Normal with unknown sigma, then:
- The sample mean Xbar is Normal with mean mu and standard deviation sigma/√n, regardless of the sample size, provided the population is Normal.
- The sample standard deviation s is an estimate of sigma, and s^2 follows a sigma^2 times a chi-square distribution with n−1 degrees of freedom, scaled by (n−1). In particular, s/ sigma has a distribution that depends on n−1 and is independent of Xbar when the population is Normal.
Thus, the quantity of interest involves the random ratio SE = s/√n divided by the true sigma/√n, which simplifies to SE/(sigma/√n) = s/sigma. Therefore, the condition |Xbar − mu| ≤ 0.6 × SE translates to:
|Xbar − mu| ≤ 0.6 × (s/√n) = 0.6 × (sigma/√n) × (s/sigma) = 0.6 × (sigma/√n) × T, where T = s/sigma.
Since Xbar − mu ~ Normal(0, sigma^2/n) and is independent of s, we can standardize the left-hand side by sigma/√n to obtain:
|Z| ≤ 0.6 × T, where Z ~ Normal(0,1) and T = s/sigma.
The distribution of T is derived from s^2 ∼ sigma^2 × (Chi-square with n−1 degrees of freedom)/(n−1). Equivalently, T = sqrt(Chi-square_{n−1} / (n−1)). In our case, n−1 = 8, so T = sqrt(Chi-square_8 / 8).
Therefore the desired probability is P(|Z| ≤ 0.6 × T) with Z ∼ N(0,1) independent of T ∼ sqrt(Chi-square_8 / 8).
To evaluate this probability, one must integrate over the joint distribution of Z and T, or equivalently compute the expectation over T of P(|Z| ≤ 0.6 × T) conditional on T. The conditional probability P(|Z| ≤ 0.6 × T | T = t) equals 2Φ(0.6 t) − 1, where Φ is the standard normal CDF. So the overall probability is E_T[2Φ(0.6T) − 1], with T = sqrt(Chi-square_8 / 8).
Carrying out this calculation (through either numerical integration or lookup of the distribution of the ratio involved, such as using a Monte Carlo approach or a closed-form integral for this setup) yields a numerical value close to 0.4349. This result reflects the combined variability from both the sampling distribution of the mean and the uncertainty in estimating sigma from the sample.
If you wanted to check the intuition without diving into the exact integral: as the degrees of freedom increase (larger n), s becomes a more precise estimator of sigma, so T tends to 1 and the probability P(|Z| ≤ 0.6 × T) tends toward P(|Z| ≤ 0.6) ≈ 2Φ(0.6) − 1 ≈ 0.7257. With n = 9 (n−1 = 8), the additional variability in s reduces the probability from that limit, yielding a result in the neighborhood of 0.43–0.44, consistent with the computed value.
In summary, the calculation combines the standard normal probability with the distribution of the sample standard deviation relative to the true sigma, and the resulting probability for n = 9 is approximately 0.4349.

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