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Use z-score boundaries to estimate normal-model percentages and counts

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An actual boundary value is given with a mean and standard deviation. What is the first calculation?

\[z=\frac{x-\mu}{s}\]
Year 12, VCE Units 3 & 4, Chapter 13, 4 difficulty levels, Calculator allowedBuild a worksheet

Estimate normal-model percentages

A normal-model percentage question may give z-score boundaries or actual boundary values. Convert actual values to signed z-scores, identify the requested region, and then use the empirical rule; scale the percentage by the population size only for an expected count

\[z=\frac{x-\mu}{s}\]
  1. Standardise an actual boundary Subtract the mean from the actual value, then divide by the standard deviation \(\displaystyle z=\frac{x-\mu}{s}\)
  2. Keep the sign A value below the mean has a negative z-score, while a value above the mean has a positive z-score \(\displaystyle x<\mu\Rightarrow z<0,\quad x>\mu\Rightarrow z>0\)
  3. Match the region Use the signed boundaries to decide whether the required values are in a central interval, one tail or two tails \(\displaystyle P(-2\le z\le2)\approx95\%,\quad P(z\ge2)\approx2.5\%\)
  4. Scale for a count Multiply the estimated percentage as a proportion by the population size only when an expected count is required \(\displaystyle \frac{95}{100}\times4000=3800\)

Worked example

The battery lives of a phone model are approximately symmetric and bell-shaped, with mean 180 hours and standard deviation 20 hours. In a population of 4000 battery lives, estimate how many are between 140 hours and 220 hours

\[\mu=180,\quad s=20,\quad 140\le X\le220,\quad n=4000\]
  1. \(\displaystyle z_{\text{lower}}=\frac{140-180}{20}=-2,\quad z_{\text{upper}}=\frac{220-180}{20}=2\)Subtract the mean from each boundary and divide by the standard deviation, retaining both signs
  2. \(\displaystyle P(-2\le z\le2)\approx95\%\)The interval is within 2 standard deviations of the mean, so about 95% of the values are in it
  3. \(\displaystyle \frac{95}{100}\times4000=3800\)Multiply the estimated proportion by 4000 to obtain an expected count of 3800

Common mistakes

Reversing the numerator
✕\(\displaystyle z=\frac{\mu-x}{s}=\frac{60-50}{5}=2\)✓\(\displaystyle z=\frac{x-\mu}{s}=\frac{50-60}{5}=-2\)
The standard-score numerator is actual value minus mean, so reversing it changes a below-mean boundary from negative to positive
Ignoring the sign
✕\(\displaystyle z=\left|\frac{50-60}{5}\right|=2\)✓\(\displaystyle z=\frac{50-60}{5}=-2\)
The sign shows whether the boundary is below or above the mean, so it must be retained
Using the complement instead of the upper tail
✕\(\displaystyle P(z\ge2)\approx97.5\%\)✓\(\displaystyle P(z\ge2)\approx2.5\%\)
A value at least 2 standard deviations above the mean is in one upper tail, so use half of the 5% outside the central 95%

Difficulty levels

introUse central regions or one boundary with integer z-values and percentage asks
standardUse one-tail or mixed-sign intervals, sometimes converting actual boundaries to z-scores
challengeUse complementary or outside regions and scale an empirical percentage to an expected count
examConvert two actual boundaries, choose the central or outside region, and give a percentage or count

Questions about this topic

Why does a value below the mean have a negative z-score?+

The z-score is (x - mean) divided by the standard deviation. If x is below the mean, x - mean is negative, so the z-score is negative

How is a one-tail percentage found from the empirical rule?+

Subtract the central percentage from 100% and divide the remainder equally between the two tails. For the central 95% interval, each tail is about 2.5%

When is an empirical-rule percentage an estimate?+

The empirical rule is an approximation for a distribution that is approximately symmetric and bell-shaped. Its percentages should not be treated as exact percentages for every data set

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 13: Normal distribution

In this chapter

Read normal parameters and describe their graph effectCalculate probabilities for normal distributionsCalculate a standard score or actual valueUse z-score boundaries to estimate normal-model percentages and countsCompare observations from different distributions using z-scoresInverse normal calculationsInfer a normal-model mean or standard deviationConditional probabilities with normal intervalsMixed probability applications with the normal distribution