Q 8.107. - Elementary Statistics

Question

Toxic Mushrooms? Refer to Exercise $8.71$

a. Determine and interpret a $99\%$ lower confidence bound for the mean cadmium level of all Boletus Pinicola mushrooms.

b. Compare your one-sided confidence interval in part (a) to the (two-sided) confidence interval found in Exercise $8.71$

Step-by-Step Solution

Answer

Part (a) The $\mu$ lower confidence bound is $0.2769$, indicating that we are $99\%$ sure that mean cadmium levels in Boletus pin cola mushrooms are less than or equal to $0.2773\mathrm{ppm}$

Part (b) The lower confidence bound of $99\%$ level (one sided) of $\mu$ is bigger than the lower confidence limit of $99\%\mathrm{Cl}$ (two sided) of $\mu$

Step 1: Given information

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

Sample size $n=12$

Part (a) Step 2: Concept

The formula used: The lower confidence bound $\overline{x}-z_{0.01}\times \frac{\sigma }{\sqrt{n}}$

Part (a) Step 3: Calculation

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

Sample size $n=12$

Confidence level $=99\%=100\times 0.99\%$

$$\begin{gathered} \therefore 1-\alpha =0.99 \\ \alpha =0.01 \\ \therefore z_{\alpha }=z_{0.01}=2.33 \end{gathered}$$

As a result, $\overline{x}-z_{0.01}\times \frac{\sigma }{\sqrt{n}}$ gives the lower confidence bound for $\mu$ of $99$ percent.

Part (a) Step 4: Calculation

Given that $\sum _{u=1}^{12}xi=6.31$

$$\begin{gathered} \therefore \overline{x}=\frac{1}{n}\sum _{u=1}^{n}xi=\frac{1}{12}\times 6.31=0.5258 \\ \therefore \overline{x}-z_{0.01}\times \frac{\sigma }{\sqrt{n}} \\ =0.5258-2.33\times \frac{0.37}{\sqrt{12}} \\ =0.5258-0.2489 \\ =0.2769 \end{gathered}$$

As a result, the $\mu$ 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 $0.2773 mathrmppm$.

Part (b) Step 1: Calculation

$99\%$ a confidence interval of mean cadmium level is $(0.2508,0.8008)$ i.e. $0.2508$ $\le \mu \le 0.8008$ and $99\%$ lower confidence bound of $\mu$ is $0.2769$ ppm i.e. $0.2769\le \mu$

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