Game plan 5 · Unit 4

The normal distribution

Probabilities, quantiles and recovering unknown parameters with z-scores.

Reach for it when

The First Question found statistics, and the question uses the phrase “normally distributed”. It will not ask you to differentiate or integrate.

Part 1

Question types

A probability is an area under the curve. A quantile works backwards from a given area to an $x$ value, $x_0$. If you are not given the mean or standard deviation, use a z-score to find it first.

Model

$$X\sim N(\mu,\sigma^{2})$$

Standardising

$$z=\frac{x-\mu}{\sigma}$$

Complex unfamiliar questions often combine a normal distribution with a binomial. Use the normal distribution to find $p$, then carry it into Game plan 3.2.

In the Topic Guides

Part 2

The three game plans

Each one starts the same way: identify the mean and standard deviation, then draw a picture. The picture is what stops you choosing the wrong tail.

Game plan 5.1

Finding a probability

  1. 1Identify the mean and standard deviation.
  2. 2Identify the given value and draw a picture.
  3. 3Use the picture to identify the upper and lower bounds.
  4. 4Solve. Use the empirical rule if Technology-free, or the NCD function if Technology-active.
P(X < 50)50μ = 55
The picture gives the bounds. Here the upper bound is $50$ and there is no lower bound, so use a very large negative number.
Game plan 5.2

Finding a quantile

  1. 1Identify the mean and standard deviation.
  2. 2Identify the given probability and draw a picture.
  3. 3Use the picture to decide whether the tail is left or right.
  4. 4Solve. Use the empirical rule if Technology-free, or the invNorm function if Technology-active.
area = 0.2tμ
The area is a left tail, so $t$ sits below the mean. For a right tail of $0.2$, the area to the left is $0.8$.
Game plan 5.3

$\mu$ or $\sigma$ unknown

  1. 1Identify the known values. The extra piece of information gives you an $x$ value and its probability.
  2. 2Calculate the z-score from that extra information.
  3. 3Substitute into the z-score formula to find the unknown mean or standard deviation.
  4. 4Answer the question in context.
known Pknown valueμ or σ = ?
The value and its probability give you $z$. Then the z-score formula has only one unknown.

In the Topic Guides

Part 3

Common QCAA language: Normal distributions

Look for “normally distributed”, “approximately normally distributed”, “modelled by a normal distribution”, $X\sim N(\mu,\sigma^{2})$ or “standard normal”. You will usually see a real-world measurement, such as height, mass, time, volume, IQ or load.

Question typeWhat the question says
Finding a probability“Determine the probability that … is between / more than / less than / at least”, “what proportion / percentage are above / below / between”, “the expected number that exceed …”. The key words are between, more than, less than, within and exceeds.Your move Mean, SD and a value are given, and you need a probability. That is normal CD: NCD(lower, upper, $\mu$, $\sigma$).
Finding a quantile“The top 15% / bottom 10%”, “taller than exactly 90% of …”, “the smallest / largest value accepted”, “the cut-off / threshold / mark needed”, “the $n$th percentile”.Your move Mean, SD and a probability are given, and you need the value. That is invNorm(area, $\mu$, $\sigma$). Sketch the curve so you shade the right tail.
$\mu$ or $\sigma$ unknown“Given [value] corresponds to [probability] … determine the mean / standard deviation”, “95% of the area lies between [two values] … determine $\sigma$”, a $z$ value given to you.Your move One of $\mu$ or $\sigma$ is unknown, but you know a value and its probability. Turn the probability into $z$, put it in $z=\frac{x-\mu}{\sigma}$ and solve.

In the Topic Guides

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