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

The prompt asks about quantifying the variability of the sample mean as a point estimate for the population mean when the sampling is a simple random sample. We begin by identifying the appropriate measure of variability for a sample mean, which is the standard error, not the standard deviation of the raw data.

Option 1: "Standard error; 0.417". This choice correctly pairs the concept (standard error) with a numerical value that represents the estimated variability of the sample mean across repeated simple random samples. The standard error reflects how much the sample mean would vary from one sample to another, given the underlying population distribution and sample size.

Option 2: "Standard deviation; 9.4". The standard deviation describes the spread of individual data points around the sample mean, not the variability of the sample mean itself. While the SD is an important measure of dispersion for the data, it does not quantify how the mean estimate would vary across samples. Thus this option misidentifies the appropriate measure for the mean’s variability.

Option 3: "Standard error; 9.4". Here the form is correct (standard error), but the numerical value 9.4 seems implausibly large for a height measurement if the context is typical human heights and a reasonable sample size. Unless the data strongly indicate such a scale, this option likely misstates the SE magnitude and should be treated as incorrect due to the inconsistent value.

Option 4: "Standard deviation; 0.417". This option again confuses the measure of dispersion (standard deviation) with the standard error, and also assigns 0.417 to the SD, which would be inconsistent with standard interpretations for height data where SD is usually larger than a fraction of a unit, depending on the dataset. As the standard deviation does not quantify the sampling variability of the mean, this option is incorrect for the question asked.

In summary, the correct concept for the variability of a sample mean under simple random sampling is the standard error, which estimates how much the mean would vary across repeated samples. The value 0.417, when valid for the SE in this context, aligns with that definition, whereas the other options either cite the wrong dispersion measure (standard deviation) for this purpose or pair the correct measure with an unlikely or inconsistent numerical value.

FA25_QTM_100_3 7.8: Heights of adults.

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