T-interval

A t-interval estimates a population mean $\mu$ (mu) as $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$, where $\bar{x}$ (x-bar) is the sample mean, $s$ is the sample standard deviation, and $t^*$ (t-star) comes from the t-distribution with $n - 1$ degrees of freedom. The t curve is there because $s$ is an estimate of the unknown $\sigma$ (sigma), and that extra uncertainty thickens the tails, making $t^*$ larger than the matching $z^*$ at every finite number of degrees of freedom.

Take 25 measurements with $\bar{x} = 12.4$ and $s = 2.1$. The standard error is $2.1/\sqrt{25} = 0.42$, the degrees of freedom are $25 - 1 = 24$, and $t^* = 2.064$ at 95% confidence. The margin of error is $2.064(0.42) = 0.8668$, so the interval runs 11.53 to 13.27.

The reading that costs marks is "95% of the measurements fall between 11.53 and 13.27." The interval estimates the mean, not the individual values. Those still scatter with $s = 2.1$, so most of them sit roughly within two standard deviations of the center, somewhere near 8.2 to 16.6, a range 4.8 times as wide as the interval. Collect more data and the gap grows rather than closes: the interval for the mean keeps shrinking while the spread of the observations stays exactly where it was.

The conditions are the three every t procedure asks for: a random sample, a sample no more than 10% of the population when sampling without replacement, and normal or large sample, met by a normal population, by $n \geq 30$, or by a graph of the data with no strong skew and no outliers. That last branch is why you plot a small sample, since one clear outlier moves both $\bar{x}$ and $s$ enough to make the interval indefensible. As the degrees of freedom grow, $t^*$ falls toward $z^*$: 2.145 at 14, 2.064 at 24, 1.984 at 99, and 1.962 at 999, against $z^* = 1.960$.

Which parameter takes which interval is covered in z-interval vs t-interval. Constructing it is Unit 4, topic 4.2, Constructing a Confidence Interval for a Population Mean or Population Mean Difference.