Infer a normal-model mean or standard deviation

Inferring normal-model parameters

Use the stated empirical-rule approximation to map rounded tail areas to signed boundaries, then solve the boundary equations

1. State the empirical approximation Map a rounded lower-tail area near 0.025 to a negative integer standard score and a rounded upper-tail area near 0.025 to a positive score \(\displaystyle \Pr(Z<z)\approx0.025\Rightarrow z\approx-2,\quad\Pr(Z>z)\approx0.025\Rightarrow z\approx2\)
2. Write the boundary equation Use the signed displacement from the mean \(\displaystyle x=\mu+zs\)
3. Use enough equations Two distinct signed standard-score boundaries give two equations when both parameters are unknown \(\displaystyle x_1=\mu+z_1s,\quad x_2=\mu+z_2s,\quad z_1\ne z_2\)
4. Check the result Substitute into both boundary equations and check that the inferred standard deviation is positive \(\displaystyle s>0\)

Worked example

Use the empirical-rule approximation for the displayed central interval of a symmetric bell-shaped reading-time model. Estimate the mean and standard deviation in seconds

1. \(\displaystyle z_1\approx-2,\quad z_2\approx2\)The empirical central interval extends about two standard deviations to each side
2. \(\displaystyle 24\approx\mu-2s,\quad40\approx\mu+2s\)Write one approximate equation for each endpoint
3. \(\displaystyle 40-24\approx4s\quad\Rightarrow\quad s\approx4\text{ seconds}\)Subtract the equations to eliminate the mean
4. \(\displaystyle \mu\approx24+2\times4=32\text{ seconds}\)Substitute the estimated spread into the lower equation
5. \(\displaystyle 32-2\times4=24,\quad32+2\times4=40\)The estimates reproduce both boundaries within the empirical model

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