Unknown mean or standard deviation
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Finding an Unknown Mean or Standard Deviation
Normal-distribution questions sometimes give a probability statement and ask for \(\mu\) or \(\sigma\).
Use standardisation
\[Z=\frac{X-\mu}{\sigma}\]
Translate the probability into a \(z\)-value, then solve the resulting equation.
Unknown mean
Suppose \(X\sim N(\mu,6^2)\) and \(P(X<72)=0.8413\).
- From standard normal tables/calculator, \(P(Z<1)=0.8413\).
- So \(\frac{72-\mu}{6}=1\).
- Therefore \(72-\mu=6\), so \(\mu=66\).
Unknown standard deviation
If \(X\sim N(100,\sigma^2)\) and \(P(X<112)=0.9772\), then \(z\approx2\).
So \(\frac{112-100}{\sigma}=2\), giving \(\sigma=6\).
Related videos
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Standard normal worked examples
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