Question

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 part per million (ppm). M. Melgar et al. measured the cadmium levels in a random sample of the edible mushroom Boletus 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 Environmental Science and Health, Vol. B33(4), pp. 439 455). Here are the data.

0.24 0.59 0.62 0.16 0.77 1.33

0.92 0.19 0.33 0.25 0.59 0.32

At the significance level, do the data provide sufficient evidence to conclude that the mean cadmium level in Boletus Pinicola mushrooms is greater than the government's recommended limit of ppm? Assume that the population standard deviation of cadmium levels in Boletus Pinicola mushrooms is 0.37 ppm.

(Note: the sum of the data is 6.31 ppm. )

Short answer

Ans: Here $z=0.24<z_{0.05}=1.645$

We do not reject $H_{0}at5\%$

The data do not provide sufficient evidence at the $5\%$ level of significance to conclude that the mean cadmium level in boletus Pinicola mushroom is greater than the government's recommended limit of $0.5$ ppm.

Step-by-step solution

Step 1. Given information.

given,

0.24 0.59 0.62 0.16 0.77 1.33

0.92 0.19 0.33 0.25 0.59 0.32

Step 2. Let μ be the mean cadmium level in the boletus Pinicola mushrooms. 

Population standard deviation, $\sigma =0.37ppm$

Now, test the hypothesis,

$\begin{array}{r}{H}_o:\mu =0.5\mathrm{ppm}\\ H_a:\mu >0.5\mathrm{ppm}\end{array}$

Level of significance, $5\%$

i.e.., $\alpha =0.05$

Step 3. Given sample size is n=12.

Sum of the data is,

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

Sample mean is,

$\begin{array}{r}\overline{x}=\frac{1}{n}\sum _{i=1}^{12} x\\ =\frac{6.31}{12}\\ =0.526\mathrm{ppm}\end{array}$

Step 4. Test statistic is,  

$\begin{array}{r}z=\frac{\overline{x}-{\mu }_0}{\frac{\sigma }{\sqrt{n}}}\\ =\frac{0.526-0.5}{\frac{0.37}{\sqrt{12}}}\\ =\frac{0.026\times \sqrt{12}}{0.37}\\ \approx 0.24\end{array}$

Step 5.  it is given that test is a right-tailed with α=0.05And from the z-table the critical value is Z0.05=1.645

Here, rejection region is,

$z>z_{0.05}=1.645$

Now, draw the graph for the above,

Step 6. Here z=0.24<z0.05=1.645

We do not reject,$H_{0}at5\%$.

Here z does not fall in the rejection region. Since the level of significance is equal to the value of the test statistic.

The data do not provide sufficient evidence at the $5\%$ level of significance to conclude that the mean cadmium level in boletus Pinicola mushroom is greater than the government's recommended limit of $0.5$ ppm.