Normal probabilities and inverse-normal values | Year 12 | Mathematical Methods

01 / LEARN

The idea, step by step

Areas in a normal model

Let Phi(z) be the standard normal lower-tail probability. Use a lower-tail calculation directly, an upper-tail complement, or subtract two lower-tail probabilities for a middle interval. Sketch the intended region to avoid choosing the wrong tail.

A normal probability is usually found numerically, not by an elementary antiderivative. State the model, bounds, mean and standard deviation so your calculation is reproducible. Keep intermediate values unrounded.

P (a < X < b) = Φ (\frac{b - μ}{σ}) - Φ (\frac{a - μ}{σ})

Turn a percentage into a boundary

An inverse-normal function needs the cumulative area to the left of the cut-off. For the highest 10%, enter a lower-tail area of 0.90, not 0.10. Then return to the original measurement scale.

A probability condition can also determine an unknown mean or standard deviation. First identify the standard normal quantile, then solve the standardisation equation. Check that a standard deviation is positive and that the direction of the resulting tail is sensible.

A common trap

Using 0.10 as the lower-tail area for the highest 10%.

Instead: The highest tenth begins at the 90th percentile.

02 / WORKED EXAMPLES

See the thinking

Example 1

An interval

$X \sim N (100, 64)$ . Find P(92<X<116).

The z-bounds are -1 and 2.
The probability is $Φ (2) - Φ (- 1)$ .
Numerically this is 0.8186.

0.8186.

Example 2

The highest tenth

A normal model has mean 52 and SD 4. Find the cut-off for the highest 10%.

Use lower-tail area 0.90.
The corresponding z-score is 1.281552.
$x = 52 + 4 z = 57.1262$ .

About 57.1262.

Check your understanding

Find the standard normal 97.5th percentile.

Show answer

1.9600.

Use inverse-normal lower-tail area 0.975.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

$X \sim N (100, 64)$ . Find P(X>112).
2 marks
A normal model has mean 50 and SD 6. Find its 2.5th percentile.
2 marks
Find the central 90% interval for a normal model with mean 52 and SD 4.
3 marks
A normal variable has SD 8 and P(X<80)=0.90. Find its mean.
3 marks
For standard normal Z, find P(Z>1 given Z>0).
2 marks
A calculator returns -0.12 for an interval probability. What should you check?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

$X \sim N (100, 64)$ . Find P(X>112).
2 marks
$z = 1.5$ ; calculate $1 - Φ (1.5)$ .
0.0668.
A normal model has mean 50 and SD 6. Find its 2.5th percentile.
2 marks
$x = 50 + 6 Φ^{- 1} (0.025)$ .
38.2402.
Find the central 90% interval for a normal model with mean 52 and SD 4.
3 marks
There is 5% in each tail.
Use $52 \pm 4 Φ^{- 1} (0.95)$ .
[45.4206, 58.5794].
A normal variable has SD 8 and P(X<80)=0.90. Find its mean.
3 marks
$(80 - μ) / 8 = Φ^{- 1} (0.90)$ .
Solve $μ = 80 - 8 Φ^{- 1} (0.90)$ .
69.7476.
For standard normal Z, find P(Z>1 given Z>0).
2 marks
Divide $P (Z > 1)$ by $P (Z > 0) = 0.5$ .
0.3173.
A calculator returns -0.12 for an interval probability. What should you check?
2 marks
Probabilities cannot be negative. Subtract the lower CDF from the upper CDF.
The calculation is invalid; check bound order and subtraction.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: continuous random variables, normal distributions and sampling distributions of sample means. Original lessons and practice.

Official curriculum source ↗

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