Inverse problems: percentiles and critical values

The table read backwards

L7.3 went from a value to a probability. The paper's favourite normal question goes the other way: the largest 10 % of invoices will be hand-reviewed — what is the threshold? Now the area is given and the value is wanted. The route is the same three steps reversed: find the $z$ whose cumulative area is the target, then un-standardise with $x=\mu +z\sigma$.

Search the body of the table for the area, and read $z$ from the margins. The target rarely appears exactly; take the nearest cell.

Worked example — the 90th percentile

Northfield's invoices are $X\sim N(120,15^2)$. The 90th percentile is the amount with 90 % of invoices below it: $\Phi (z)=0.90$. In the body of the table 0.8997 sits at $z=1.28$ and 0.9015 at $z=1.29$; 0.8997 is nearer, so $z=1.28$ and

$$x=120+1.28\times 15=120+19.2=\$139.20$$

Ten percent of invoices exceed \$139.20. (A calculator's exact $z$ is 1.2816, giving \$139.22; the paper's tolerance absorbs the difference, and the cell value is what the marking scheme expects.)

The lowest 5 % — and the 1.645 rule

The cutoff below which the smallest 5 % of invoices fall has $\Phi (z)=0.05$. The two nearest cells are 0.0505 at $z=-1.64$ and 0.0495 at $z=-1.65$ — the target sits exactly between them. The convention, memorised: take the midpoint, $z=-1.645$. Then

$$x=120-1.645\times 15=120-24.675=\$95.33$$

The same number, positive, is $z_{0.05}=1.645$ — the value with 5 % of the area above it, and the critical value of every one-sided 5 % test from Module 10 on.

A middle percentage: halve the tails first

Between what two amounts do the middle 80 % of invoices lie? The middle 80 % leaves 20 % in the tails — 10 % in each. The lower limit has $\Phi (z)=0.10$, so $z=-1.28$; the upper has $\Phi (z)=0.90$, so $z=1.28$:

$$120-1.28(15)=\$100.80\text{and}120+1.28(15)=\$139.20$$

The error to avoid is looking up 0.80 directly — that gives $z=0.84$, the 80th percentile, a different question. For a middle $1-\alpha$, the $z$ comes from $\Phi (z)=1-\alpha /2$.

The three critical values

The confidence intervals of Modules 9 and 12 use the middle-percentage $z$ constantly, and three of them are to be memorised:

The subscript is the upper-tail area. $z_{0.05}=1.645$ serves both the 90 % interval and the one-sided 5 % test; $z_{0.025}=1.96$ serves the 95 % interval and the two-sided 5 % test. A fourth, $z_{0.10}=1.28$, is the one this lesson's percentile used.