The probability \(P\) that a measurement will fall within \(t\) standard deviations is given by \(P=\int_{\mu-t \sigma}^{\mu+t \sigma} \frac{\exp \left(\frac{-(x-\mu)^{2}}{2 \sigma^{2}}ight)}{\sigma \sqrt{2 \pi}} d x\).

(a) Show how this probability expression reduces to the normal error integral \(P=\) \(\frac{1}{\sqrt{2 \pi}} \int_{-t}^{t} e^{-\frac{z^{2}}{2}} d z\)

(b) Using the normal error integral and the Gaussian integration formula (for \(n=\) 2), calculate the probability that a measurement will fall within one standard deviation (that is, \(t=1\) ). What is the relative error for this estimation?