91Ó°ÊÓ

To illustrate the preceding relationship by obtaining the appropriate upper confidence bound and comparing the result to the conclusion of the hypothesis test in the specified exercise $9.85$.

Since, the population standard deviation as $\mathrm{σ}=4.2\mathrm{mg}$

The null hypothesis is determined as follows:
$H_0:\mathrm{μ}={\mathrm{μ}}_0$

$H_0:\mathrm{μ}=18\mathrm{mg}$

$H_a:\mathrm{μ}<18\mathrm{mg}$

Sample size,$n=45$.
Sample mean, $\stackrel{¯}{x}=16.68$
Calculate the population mean's $100(1-\mathrm{Î\pm })\%$ upper confidence bound (one sided one mean z-interval) using MINITAB.
The population mean's $99\%$upper confidence bound is $16.137mg$.
The null hypothesis mean $\mathrm{μ}=18mg$ is higher than the upper confidence bound obtained.
It shows that we have rejected the null hypothesis $H_0:\mathrm{μ}=18\mathrm{mg}$in favor of the alternative hypothesis $H_a:\mathrm{μ}<18\mathrm{mg}$at the $1\%$ level of significance.
As a result, the null hypothesis $H_0:\mathrm{μ}=18\mathrm{mg}$ is rejected in support of the alternative hypothesis $H_a:\mathrm{μ}<18\mathrm{mg}$.

To illustrate the preceding relationship by obtaining the appropriate upper confidence bound and comparing the result to the conclusion of the hypothesis test in the specified exercise $9.86$.

Since, the population standard deviation $\mathrm{σ}=6.8$years .

The null hypothesis is determined as follows:

$H_0:\mathrm{μ}={\mathrm{μ}}_0$

$H_0:\mathrm{μ}=55$years.

And the alternative hypothesis is determined as follows:

$H_a:\mathrm{μ}<{\mathrm{μ}}_0$

$H_a:\mathrm{μ}<55$years.

And the significance level $=1\%$i.e. $\mathrm{Î\pm }=0.01$.
Then the sample size, $n=21$.
The sample mean, $\stackrel{¯}{x}=52.5$
Calculate the population mean's $100(1-\mathrm{Î\pm })$ upper confidence bound (one sided one mean z-interval) using MINITAB.
The population mean has a $99\%$ upper confidence bound of $55.95$ years.
The null hypothesis mean $H_0:\mathrm{μ}=55$ years is lower than the upper confidence bound obtained.
It shows that we have not rejected the null hypothesis $H_0:\mathrm{μ}=55$years in support of the alternative hypothesis $H_a:\mathrm{μ}=55$years at the $1$ percent level of significance.
As a result, the null hypothesis $H_0:\mathrm{μ}=55$years is not rejected in support of the alternative hypothesis $H_a:\mathrm{μ}<55$ years.