How do you find a normal probability with technology?

For X distributed as N of mu comma sigma squared, a probability is an area under the bell curve. Use the calculator command normalcdf with a lower bound, an upper bound, the mean mu and the standard deviation sigma. For P of X less than a use a very large negative lower bound; for P of X greater than a use a very large positive upper bound; for P of a less than X less than b enter both a and b.

What is the standard normal distribution?

The standard normal is the normal distribution with mean 0 and standard deviation 1, written Z distributed as N of 0 comma 1. Any normal value can be standardised to a z-score with z equals x minus mu over sigma, and then P of X less than a equals P of Z less than that z-score.

How do you use symmetry to find a normal probability?

The normal curve is symmetric about the mean. So P of X greater than mu equals 0.5, the upper tail P of Z greater than z equals 1 minus P of Z less than z, and by symmetry P of Z less than minus z equals 1 minus P of Z less than z.

What is the inverse normal used for?

The inverse normal, invNorm, goes the other way: given a probability it returns the value. To find x with P of X less than x equal to p, use invNorm of p comma mu comma sigma. This gives quantiles such as percentiles and the ends of a central interval.

How do you find a value for a top percentage?

A top 10 percent cutoff has P of X greater than x equal to 0.10, which means P of X less than x equal to 0.90. So x equals invNorm of 0.90 comma mu comma sigma. Always convert an upper-tail probability to a lower-tail probability before using invNorm.

Do you enter the variance or the standard deviation?

The notation N of mu comma sigma squared states the variance, but normalcdf and invNorm both take the standard deviation sigma. For X distributed as N of 100 comma 36 the standard deviation is the square root of 36, which is 6.

How do you find a normal probability with technology?

For X distributed as N of mu comma sigma squared, use normalcdf with a lower bound, an upper bound, the mean mu and the standard deviation sigma. For P of X less than a use a large negative lower bound; for P of X greater than a use a large positive upper bound; for an interval enter both a and b.

What is the standard normal distribution?

The normal distribution with mean 0 and standard deviation 1, written Z distributed as N of 0 comma 1. Standardise with z equals x minus mu over sigma, then P of X less than a equals P of Z less than that z-score.

How do you use symmetry to find a normal probability?

The curve is symmetric about the mean, so P of X greater than mu equals 0.5, P of Z greater than z equals 1 minus P of Z less than z, and P of Z less than minus z equals 1 minus P of Z less than z.

What is the inverse normal used for?

invNorm returns a value from a probability. For x with P of X less than x equal to p, use invNorm of p comma mu comma sigma. It gives percentiles and the ends of central intervals.

Do you enter the variance or the standard deviation?

N of mu comma sigma squared states the variance, but normalcdf and invNorm take the standard deviation sigma. For N of 100 comma 36 use sigma equals 6.

Year 12 Methods (Unit 3 & 4) The normal distribution

Determining normal probabilities

20 practice questions 0 video lessons Theory + worked examples

In Year 12 Mathematical Methods (Queensland, QCAA), a normal probability is an area under the bell curve of \(X\sim N(\mu,\sigma^2)\). You find it with technology — the command \(\operatorname{normalcdf}(\text{lower},\text{upper},\mu,\sigma)\) — or by standardising to \(Z\sim N(0,1)\) and reading \(P(Zinverse normal \(\operatorname{invNorm}\) does the reverse: it returns the value \(x\) for a given probability. Unit 4 Topic 3, Continuous random variables and the normal distribution.

A normal variable is written \(X\sim N(\mu,\sigma^2)\), with mean \(\mu\) and standard deviation \(\sigma\). Its graph is the symmetric bell curve, and the total area under it is \(1\). A probability such as \(P(aarea of the shaded strip between \(x=a\) and \(x=b\).

Using technology. Queensland uses a calculator (distribution tables are not required). The command \(\operatorname{normalcdf}(\text{lower},\text{upper},\mu,\sigma)\) returns the area between the two bounds. For a lower tail \(P(Xa)\) use a very large positive upper bound (\(\infty\)).

The standard normal. The standard normal \(Z\sim N(0,1)\) has mean \(0\) and standard deviation \(1\). Every normal value can be standardised to a z-score \(z=\dfrac{x-\mu}{\sigma}\), and then \(P(X

Symmetry. Because the curve is symmetric about \(\mu\), the mean splits the area in half: \(P(X>\mu)=P(X<\mu)=0.5\). For any value, \(P(X>a)=1-P(X

The inverse normal (quantiles). \(\operatorname{invNorm}(p,\mu,\sigma)\) returns the value \(x\) with \(P(Xquantiles — percentiles, the top or bottom \(k\%\) cutoff, and the ends of a central interval.

Key idea. A probability turns a value into an area (use \(\operatorname{normalcdf}\)); the inverse normal turns an area back into a value (use \(\operatorname{invNorm}\)). Sketch the shaded region first — it tells you which command and which tail you need.

\(P(a

\(x=\operatorname{invNorm}(0.90,\mu,\sigma)\): the value with \(0.90\) of the area to its left

A lower-tail probability with technology:

\[P(X

P (X < a)

An interval probability:

\[P(a

Standardise, then read the standard normal:

\[z=\dfrac{x-\mu}{\sigma}\qquad P(X

z = \frac{x - μ}{σ}

Symmetry and the complement:

\[P(X>a)=1-P(X\mu)=0.5,\qquad P(Z<-z)=1-P(Z

The inverse normal (a quantile from a probability):

\[x=\operatorname{invNorm}(p,\,\mu,\,\sigma)\quad\text{gives } x \text{ with } P(X

x = invNorm (p, μ, σ)

Enter \(\sigma\), not \(\sigma^2\). The notation \(N(\mu,\sigma^2)\) states the variance, but \(\operatorname{normalcdf}\) and \(\operatorname{invNorm}\) both take the standard deviation \(\sigma\). Give probabilities to \(4\) decimal places.

How to determine a normal probability

Write down \(\mu\) and \(\sigma\). From \(N(\mu,\sigma^2)\) take \(\sigma=\sqrt{\text{variance}}\).
Sketch and shade. Draw the bell, mark the mean, and shade the region asked for. This shows whether it is a lower tail, an upper tail, or an interval.
Choose the command. Area between values → \(\operatorname{normalcdf}(\text{lower},\text{upper},\mu,\sigma)\). A value from an area → \(\operatorname{invNorm}(p,\mu,\sigma)\).
Use symmetry when it helps. \(P(X>\mu)=0.5\); \(P(X>a)=1-P(Xx)=k\) into \(P(X
Round sensibly. Probabilities to \(4\) decimal places; a value (quantile) to the accuracy of the context.

Standardising is optional but handy. You may enter \(\mu\) and \(\sigma\) straight into \(\operatorname{normalcdf}\), or first standardise with \(z=\dfrac{x-\mu}{\sigma}\) and use \(P(Z
    

Example 1 — A lower-tail probability

\(X\sim N(60,\,5^2)\). Find \(P(X<66)\) to \(4\) decimal places.

Solution

Enter \(\mu=60,\ \sigma=5\) into \(\operatorname{normalcdf}\).

\(P(X<66)\)	\(=\)	\(\operatorname{normalcdf}(-\infty,66,60,5)\)
\(\)	\(=\)	\(0.8849\)

0.8849

Example 2 — An upper tail by symmetry

For \(X\sim N(25,\,4^2)\), find \(P(X>30)\) to \(4\) decimal places.

Solution

An upper tail is the complement of the lower tail.

\(P(X>30)\)	\(=\)	\(1-P(X<30)\)
\(\)	\(=\)	\(1-0.8944\)
\(\)	\(=\)	\(0.1056\)

(or \(\operatorname{normalcdf}(30,\infty,25,4)=0.1056\) directly).

0.1056

Example 3 — Standardise for an interval

\(X\sim N(40,\,5^2)\). Find \(P(33

Solution

Standardise both ends, then read the standard normal.

\(z_1\)	\(=\)	\(\dfrac{33-40}{5}=-1.4\)
\(z_2\)	\(=\)	\(\dfrac{47-40}{5}=1.4\)
\(P(33	\(=\)	\(P(-1.4

0.8385

Example 4 — Inverse normal (a quantile)

Battery life is \(X\sim N(200,\,20^2)\) hours. Find the life below which \(90\%\) of batteries fall.

Solution

Use \(\operatorname{invNorm}\) with the lower-tail probability \(0.90\).

\(x\)	\(=\)	\(\operatorname{invNorm}(0.90,200,20)\)
\(\)	\(=\)	\(225.63\ \text{hours}\)

225.63

Common pitfalls

Shade the right tail. \(P(X>a)\) is the area to the right of \(a\); \(P(X

Enter \(\sigma\), not \(\sigma^2\). \(N(\mu,\sigma^2)\) states the variance, but \(\operatorname{normalcdf}\) and \(\operatorname{invNorm}\) take the standard deviation. For \(N(100,36)\) use \(\sigma=6\).

Convert a top \(k\%\) before \(\operatorname{invNorm}\). A top \(10\%\) cutoff means \(P(X>x)=0.10\), i.e. \(P(X

Keep the sign of \(z\). A value below the mean has a negative z-score; dropping the sign moves the probability to the wrong side of the curve.

Frequently asked questions

Which calculator command do I use?

\(\operatorname{normalcdf}(\text{lower},\text{upper},\mu,\sigma)\) for a probability (an area), and \(\operatorname{invNorm}(p,\mu,\sigma)\) for a value from a lower-tail probability \(p\).

How do I do \(P(X>a)\)?

Either \(\operatorname{normalcdf}(a,\infty,\mu,\sigma)\), or \(1-P(X

Do I have to standardise?

No. You can enter \(\mu\) and \(\sigma\) directly. Standardising with \(z=\dfrac{x-\mu}{\sigma}\) is useful when a question is phrased in z-scores or asks you to show the working.

What is a quantile?

The value with a given proportion below it. The \(90\)th percentile has \(P(X

How do I find a central interval?

A central \(90\%\) leaves \(5\%\) in each tail, so its ends are \(\operatorname{invNorm}(0.05,\mu,\sigma)\) and \(\operatorname{invNorm}(0.95,\mu,\sigma)\).

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

← Previous subtopic

Standardisation

Next subtopic →

Solving problems using the normal distribution