Critical Value vs T-Distribution

A critical value is a quantile of a reference distribution, chosen so that a stated area falls beyond it. For a confidence interval at level $C$ it is the number $z^*$ (z-star) or $t^*$ (t-star) with the middle $C$ of the curve lying between $-z^*$ and $z^*$. For a significance test it is the point cutting off $\alpha$ (alpha) in whichever tail the alternative hypothesis points to. One idea, two jobs: it sets a width, or it sets a boundary.

The standard normal values, listed in full on the z-table:

At 95%, 1.960 is the value leaving 0.025 in each tail, so 0.95 of the standard normal curve sits between $-1.960$ and $1.960$. With the population standard deviation unknown you switch to $t^*$ at the right degrees of freedom, and it is larger than the matching $z^*$ at every finite df. At 95% the t-table gives 2.262 at 9 df, 2.064 at 24 df, and 1.984 at 99 df, closing on 1.960 from above without reaching it.

The error worth naming is "the critical value for 95% confidence is 1.96." It is 1.96 only when the reference curve is the standard normal. A mean from a sample of 12 uses $t$ with 11 df, where the critical value is 2.201, and with a normal population, reaching out only 1.96 standard errors captures the true mean 92.4% of the time rather than 95%. The interval is about 11% too narrow and it undercovers by more than two points.

Not every critical value comes in a $\pm$ pair. The chi-square distribution is right skewed and its tests use the upper tail alone, so there is a single cutoff: 7.815 at 3 degrees of freedom and $\alpha = 0.05$, from the chi-square table. A critical value also marks only the boundary. How far past it you landed is what a p-value reports, which is why two tests can share a critical value and carry very different evidence.

The $t$-distribution is not one curve but a family, indexed by the degrees of freedom. It is the distribution of $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ (x-bar minus mu, over s divided by root n) when the data come from a normal population. Swapping the fixed $\sigma$ (sigma) for the sample standard deviation $s$, which itself changes from sample to sample, is what puts the extra weight in the tails. Every member is symmetric about 0, and the family closes on the standard normal as the degrees of freedom grow.

The numbers make that convergence concrete. For a 95 percent interval the t-table gives $t^* = 2.145$ at 14 degrees of freedom, $2.042$ at 30, $1.984$ at 100 and $1.962$ at 1000, against $z^* = 1.960$ for the normal. The gap is 8.6 percent of the critical value at 14 degrees of freedom and about 0.1 percent at 1000.

"The sample is small so use $t$, and large so use $z$" is the wrong rule, and it is the one most students carry in. The trigger is whether $\sigma$ is known, not how big $n$ is. With 500 observations and a standard deviation estimated from them, the correct model is $t$ on 499 degrees of freedom, which happens to sit very close to the normal. Knowing $\sigma$ with $n = 8$ would put you back on $z$.

The heavier tails change verdicts, not just widths. A statistic of 2.00 read on $t$ with 14 degrees of freedom has a two-sided p-value of 0.0653, against the 0.0455 the normal returns for the same 2.00, so at $\alpha = 0.05$ one model rejects and the other does not.

Those tails cover the uncertainty in $s$ and nothing else. They do not repair a skewed population or a stray outlier, which is why a $t$ procedure still asks you to look at the shape of the sample first. The $t$-distribution enters the course at topic 4.2 of Unit 4, Inference for Quantitative Data: Means.