When you do not know the variance, and need to derive it from the data, which distribution do you use to determine the probability of a mean?

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Question restated: When you do not know the variance, and need to derive it from the data, which distribution do you use to determine the probability of a mean?

First, notice there are no answer choices provided in the data. In standard statistical practice, the appropriate distribution for constructing confidence intervals or testing hypotheses about a population mean when the population variance is unknown and the sample size is not extremely large is the t-distribution. This is because the sample standard deviation, used as an estimate of the true standard deviation, adds extra variability to the estimate of the standard error of the mean.

Key idea: if the population variance is known, you would typically use the normal (z) distribution to model the sampling distribution of the mean. However, when the variance is unknown and replaced with the sample variance, the sampling distribution of the sample mean divided by the estimated standard error follows a t-distribution with df = n - 1.

Reasoning about why t is used:
- The standard error of the sample mean is computed as s / sqrt(n), where s is the sample standard deviation. Since s is a random variable, the standardized mean involves both the sample mean and s, which changes the distribution from normal to t.
- The t-distribution accommodates the additional uncertainty from estimating the variance, especially for small to moderate sample sizes.
- As the sample size grows very large, the t-distribution converges to the normal distribution, which is why z-based methods become adequate in large samples.

Why other interpretations would be incorrect (conceptual contrasts):
- Using a normal distribution would ignore the extra variability introduced by estimating the variance, leading to underestimated tails and overconfident conclusions in small samples.
- Some might think of using a chi-square distribution for variance itself, but that pertains to the distribution of a sample variance estimator, not directly to the distribution of the sample mean when the variance is unknown.
- If the variance were known, the appropriate model would be the normal distribution for the mean, not the t-distribution.

In summary, when variance is unknown and derived from the data, the standard approach is to use the t-distribution to model the probability of the sample mean, reflecting the additional uncertainty from estimating the variance.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 7. Quiz for Class 7

When you do not know the variance, and need to derive it from the data, which distribution do you use to determine the probability of a mean?

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