The sample means obtained are point estimates for the mean height of all active individuals, if the sample of individuals is equivalent to a simple random sample. What measure do we use to quantify the variability of such an estimate? Compute this quantity using the data from the original sample under the condition that the data are a simple random sample.

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BlackTom AI · Accepted Answer

To approach this question, first recognize what we are quantifying: the variability of the estimate of a population mean when using a sample. That kind of variability is described by the standard error of the mean, not by the standard deviation of the individual observations.
Option A suggests the standard deviation with a value of 9.4. The standard deviation measures how dispersed individual heights are around the sample mean, not how much the sample mean itself would vary from sample to sample. Since the question asks about the variability of the estimate of the mean, this option mismatches the concept entirely, and the numeric value does not address the sampling distribution of the mean.
Option B states the standard error with a value of 0.417. The standard error (SE) quantifies the expected fluctuation of the sample mean if we were to repeatedly draw SRS samples from the population. This aligns with the idea of a point estimate (the sample mean) having sampling variability. If the data given yield SE = 0.417 under an SRS assumption, this is the appropriate measure and number to report.
Option C proposes the standard deviation with a value of 0.417. As noted earlier, the SD measures spread of individual data points, not the precision of the mean as an estimator. A SD of 0.417 would imply extremely tight dispersion among heights, which is unlikely for a variable like height and also misidentifies the quantity of interest.
Option D combines the standard error with a value of 9.4. While it correctly labels the measure as SE, the numeric value conflicts with the typical scale of SE for sample means unless the sample size is extremely small or the data are mis-scaled. In standard practice, an SE as large as 9.4 would indicate enormous sampling variability, inconsistent with the premise of a typical height sample and with the provided target value in option B.
In summary, the concept you should apply is that the standard error of the mean is the right descriptor of the sampling variability of the estimate, whereas the standard deviation describes the spread of individual observations. The only option that correctly pairs the right measure with a plausible, correctly oriented value is Option B: Standard error; 0.417.

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