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The null hypothesis, denoted by H₀, is the claim that is initially assumed to be true (the 鈥減rior belief鈥� claim). The alternative hypothesis, denoted by H_a, is the assertion that is contradictory to H₀.

The null hypothesis will be rejected in favour of the alternative hypothesis only if sample evidence suggests that H₀ is false. If the sample does not strongly contradict H₀, we will continue to believe in the plausibility of the null hypothesis. The two possible conclusions from a hypothesis-testing analysis are then reject H0 or fail to reject H₀.

The z values are taken from the appendix table A.3 as well as \(\Phi (z)\) values.

Testing \({H_0}:\mu = 95\) versus \({H_a}:\mu \ne 95\) based on a sample of size \(n = 16\), which is from a normal population with standard deviation \(\sigma = 1.2\).

Assumption that the sample is from normal population distribution with the known standard deviation allows using the test statistic,

\(Z = \frac{{\overline X - {\mu _0}}}{{\sigma /\sqrt n }}\)

The value of the statistic, for \(\overline x = 94.32\), the test statistic value is,

\(\begin{array}{l}z = \frac{{\overline x - {\mu _0}}}{{\sigma /\sqrt n }}\\z = \frac{{94.32 - 95}}{{1.2/\sqrt {16} }}\\z = - 2.27\end{array}\)

The P value for the two sides alternative hypothesis is,

\(\begin{array}{l}{\rm{P(Z > - z and Z < z) = P(Z > 2}}{\rm{.27 and Z < - 2}}{\rm{.27)}}\\{\rm{ = 2}}\Phi {\rm{( - 2}}{\rm{.27)}}\\{\rm{ = 2(0}}{\rm{.0116)}}\\{\rm{ = 0}}{\rm{.0232}}\end{array}\)

The P value\({\rm{0}}{\rm{.0232}}\) is bigger than \(\alpha {\rm{ = 0}}{\rm{.01}}\), do not reject hypothesis \({H_0}\).

Using the two-tailed level \(0.01\) test, then

\(\begin{array}{l}{z_{\alpha /2}} = {z_{0.005}}\\ = 2.58\end{array}\)

Type II error probability \(\beta (\mu ')\) for a level \(\alpha \) test, when alternative hypothesis is \({H_a}:\mu \ne {\mu _0}\) is ,

\(\beta (\mu ') = \Phi \left( {{z_{\alpha /2}} + \frac{{{\mu _0} - \mu '}}{{\sigma /\sqrt n }}} \right) - \Phi \left( { - {z_{\alpha /2}} - \frac{{{\mu _0} - \mu '}}{{\sigma /\sqrt n }}} \right)\)

The type II error when \(\mu ' = 94\) is,

\(\begin{array}{l}\beta (94) = \Phi \left( {2.58 + \frac{{95 - 94}}{{1.2/\sqrt {16} }}} \right) - \Phi \left( { - 2.58 - \frac{{95 - 94}}{{1.2/\sqrt {16} }}} \right)\\ = \Phi (5.91) - \Phi (0.75)\\ = 0.9999 - 0.7733\\\beta (94) = 0.2266\end{array}\)

The required sample size \(n\) for which a level \(\alpha \) test produces \(\beta (\mu ') = \beta \) for upper or lower test is,

\(n = {\left( {\frac{{\sigma ({z_\alpha } + {z_\beta })}}{{{\mu _0} - \mu '}}} \right)^2}\)

From the appendix or a software for \({z_{\alpha /2}} = 2.58\) and for \(\beta = 0.01,{z_\beta } = 1.28\)

Hence the necessary sample size is

\(\begin{array}{l}n = {\left( {\frac{{1.2(2.58 + 1.28)}}{{95 - 94}}} \right)^2}\\n = 21.46\end{array}\)

But the integer is needed, round it up to \(n = 22\).