Questions: For the normal distribution, the mean plus and minus two standard deviations will include about what percent of the observations?

Solution Steps

We consider a normal distribution with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). The sample size is \( n = 1 \).

We calculate the range for the mean plus and minus two standard deviations: \[ \text{Range Start} = \mu - 2\sigma = 0 - 2 \cdot 1 = -2 \] \[ \text{Range End} = \mu + 2\sigma = 0 + 2 \cdot 1 = 2 \]

The Z-scores corresponding to the range are: \[ Z_{\text{start}} = \frac{-2 - \mu}{\sigma} = \frac{-2 - 0}{1} = -2.0 \] \[ Z_{\text{end}} = \frac{2 - \mu}{\sigma} = \frac{2 - 0}{1} = 2.0 \]

Using the cumulative distribution function \( \Phi \), we find the probability that the sample mean falls within the specified range: \[ P = \Phi(Z_{\text{end}}) - \Phi(Z_{\text{start}}) = \Phi(2.0) - \Phi(-2.0) = 0.9545 \]

Final Answer