The t-Distribution and Small Sample Inference

When the population variance $\sigma^2$σ2 is unknown and the sample size is small, we cannot use the normal distribution directly for inference about the mean. Instead, we use the t-distribution, which accounts for the additional uncertainty from estimating $\sigma$σ with the sample standard deviation $s$s.

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Why the t-Distribution?

When $\sigma$σ is known:

$Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \sim N(0, 1)$Z=σ/n​Xˉ−μ​∼N(0,1)

When $\sigma$σ is unknown and replaced by the sample standard deviation $s$s:

$T = \frac{\bar{X} - \mu}{s / \sqrt{n}} \sim t_{n-1}$T=s/n​Xˉ−μ​∼tn−1​

The statistic $T$T follows a t-distribution with $n - 1$n−1 degrees of freedom, not a standard normal. This is because $s$s is a random variable that introduces extra variability.

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Properties of the t-Distribution

Property	Detail
Shape	Symmetric, bell-shaped (similar to the normal)
Centre	Mean = 0
Tails	Heavier tails than the normal (more probability in the tails)
Parameter	Degrees of freedom $\nu = n - 1$ν=n−1
As $\nu \to \infty$ν→∞	The t-distribution approaches $N(0, 1)$N(0,1)