91Ó°ÊÓ

Population SD $\mathrm{σ}=0.37\mathrm{ppm}$

Sample size $n=12$

The formula used: The lower confidence bound $\stackrel{¯}{x}-z_{0.01}×\frac{\mathrm{σ}}{\sqrt{n}}$

The lower confidence bound for the population mean calcium level $\mathrm{μ}$Population SD $\mathrm{σ}=0.37\mathrm{ppm}$must be determined.

Sample size $n=12$

Confidence level $=99\%=100×0.99\%$

$$\begin{gathered} ∴1-\mathrm{Î\pm }=0.99 \\ \mathrm{Î\pm }=0.01 \\ ∴z_{\mathrm{Î\pm }}=z_{0.01}=2.33 \end{gathered}$$

As a result, $\stackrel{¯}{x}-z_{0.01}×\frac{\mathrm{σ}}{\sqrt{n}}$ gives the lower confidence bound for $\mathrm{μ}$of $99$percent.

Given that $\underset{u=1}{\overset{12}{∑}}xi=6.31$

$$\begin{gathered} ∴\stackrel{¯}{x}=\frac{1}{n}\underset{u=1}{\overset{n}{∑}}xi=\frac{1}{12}×6.31=0.5258 \\ ∴\stackrel{¯}{x}-z_{0.01}×\frac{\mathrm{σ}}{\sqrt{n}} \\ =0.5258-2.33×\frac{0.37}{\sqrt{12}} \\ =0.5258-0.2489 \\ =0.2769 \end{gathered}$$

As a result, the $\mathrm{μ}$lower confidence bound is $0.2769$, indicating that we are $99$ percent sure that mean cadmium levels in Boletus pin cola mushrooms are less than or equal to $0.2773 mathrmppm$.

$99\%$a confidence interval of mean cadmium level is $(0.2508,0.8008)$i.e. $0.2508$$≤\mathrm{μ}≤0.8008$and $99\%$lower confidence bound of $\mathrm{μ}$is $0.2769$ppm i.e. $0.2769≤\mathrm{μ}$

We can see from the above two results that the lower confidence bound of $99\%$ level (one sided) of $\mathrm{μ}$ is bigger than the lower confidence limit of $99\%\mathrm{Cl}$ (two sided) of $\mathrm{μ}$