Statistical Tables

This appendix collects the two tables MATH 1209 needs all semester: the standard Normal (z) table (Ch 4–7) and the Student t table (Ch 7). Both answer the same kind of question two different ways — “how much area?” or “what cutoff value?” — and both have a calculator shortcut and an R shortcut, so you never have to read the printed grid unless you want to check your work by hand. Every value below was computed directly (not copied from another book), so it matches exactly what invNorm, normalcdf, qnorm, and qt return on your own calculator or in R.

11. The standard Normal (z) table¶

The z-table gives the area to the left of a z-score under the standard Normal curve $N(0,1)$ — that is, $P(Z < z)$. Find the whole-number-and-tenths part of your z-score down the left column, then the hundredths part across the top row; the cell where they meet is the area.

Calculator + R equivalents. Any value in this table is one keystroke away — prefer these over reading the grid whenever you have a device in hand.

22. The Student t table¶

The t-distribution is used instead of the Normal whenever you estimate a mean’s standard error from the sample standard deviation $s$ rather than knowing the true $\sigma$ — which is the everyday case in Ch 7. It is bell-shaped and centered at 0 like the Normal, but with heavier tails; its exact shape depends on the degrees of freedom, $df = n - 1$ for one-sample work. Unlike the z-table, the t-table does not print every possible area — it prints only the critical values ($t^{*}$) you need for common confidence levels and significance levels, because that is what a confidence interval or a hypothesis-test decision actually requires.

Notice the bottom row: as $df$ grows, every column in the t-table slides down toward the matching z-critical value from the Course Formula Sheet (1.645 / 1.960 / 2.576 for 90% / 95% / 99%). That is the t-distribution’s heavier tails fading away as the sample gives you more information about $\sigma$ — with a huge sample, “not knowing $\sigma$” barely costs you anything.

Calculator + R equivalents.

In practice, the TInterval, T-Test, t.test(), and tsum.test() procedures on the Course Formula Sheet and the TI-83/84 Quick Guide already look up $t^{*}$ for you — this table exists so you can see where that number came from and check it by hand.