Toxic Mushrooms?Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at $0.5$part per million (ppm). M. Melgar et al. measured the cadmium levels in a random sample of the edible mushroom Bolefus pinicola and published the results in the paper "Influence of Some Factors in Toxicity and Accumulation of Cd from Edible Wild Macrofungi in NW Spain (Journal of Envionmental Science and Health. Here are the data obtained by the researchers.

Find and interpret a $99\%$confidence interval for the mean cadmium level of all Boletus pinicola mushrooms. Assume a population standard deviation of cadmium levels in Bolefus pinicola mushrooms of $0.37\mathrm{ppm}$.

Step by step solution

01

Given Information

Population standard deviation is $0.37$.

Data of twelve individuals are given:

02

Determination of sample mean

We are given that

$\sum _{i=1}^{12}{x}_i=6.31$

Hence, sample mean$\overline{x}=\frac{1}{n}\times \sum _{i=1}^{n}xi=\frac{1}{12}\times 6.31=0.5258$

03

Determination of confidence interval

$99\%$confidence level of $\mu$$=100\times 0.99\%$

$\Rightarrow 1-\alpha =0.99$

role="math" localid="1652722332964" $\Rightarrow \alpha =0.01$

$99\%$confidence level of $\mu$is calculated as

$\overline{x}-z_{0.005}\times \frac{\sigma }{\sqrt{n}},\overline{x}+z_{005}\times \frac{\sigma }{\sqrt{n}}$

Now,

role="math" localid="1652722501303" $x-z_{0.005}\times \frac{\sigma }{\sqrt{n}}=0.5258-2.575\times \frac{0.37}{\sqrt{12}}$

role="math" localid="1652722561196" $=0.5258-0.275$

$=0.2508$

Also,

role="math" localid="1652722574315" $\overline{x}+z_{0.005}\times \frac{\sigma }{\sqrt{n}}=0.5258+0.275$

$=0.8008$

Hence, confidence interval of$\mu$is$(0.2508,0.8008)$

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