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Chapter 9: Q10E (page 548)

In Exercise 8, assume that Z=z is observed. Find a formula for thep-value.

Short Answer

Expert verified

The formula for the p-value is1- 筛(Z)

Step by step solution

01

Step 1:Definition of the p-value

A p-value, by definition, is the supremum of the probability of obtaining the value of the statistic as extreme as observed if H0 is true

In the given context,X1,鈥�.,Xn~N (碌,1) and

Z = 鈭歯 (x虅n- 碌0)~N (碌 - 碌0 ,1)

02

Step 2:Derivation of the p-value

The null hypothesis is

H0: 碌 鈮� 碌0

thus, on observing Z=z, the p-value is

\begin{aligned}sup\underset{_{H0}}{}P(Z\geq z)=sup\underset{\mu\leq _\mu{0}}{}(Z\geq z)\end{aligned}

\begin{aligned}=sup\underset{_{\mu0\leq\mu}}{}P(Z-\mu +\_\mu{0}\geq z-\mu+\mu_{0})\end{aligned}

\begin{aligned}=sup\underset{\mu\leq\mu_{0}}{}1-\Phi(z-\mu+{\mu 0})=1-\Phi (z)\end{aligned}

The formula for the p-value is \begin{aligned}1-\Phi(z)\end{aligned}

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