Question

Studies indicate that drinking water supplied by some old lead-lined city piping systems may contain harmful levels of lead. Based on data presented by Karalekas and colleagues,it appears that the distribution of lead content readings for individual water specimens has mean \(.033 mg/L\) and standard deviation \(.10 mg/L\). Explain why it is obvious that the lead content readings are not normally distributed.

Answer 1

The mean lead content reading in drinking water supplied by the old piping systems in the city involving lead-lined pipes is, \(\mu = 0.033mg/L\) and the standard deviation is \(\sigma = 0.10mg/L\).
In case the reading is zero, it would mean that the particular specimen of drinking water does not have any trace of lead in it. However, a negative reading is meaningless and impossible.
According to the empirical rule, if a random variable follows normal distribution, then about 68% of the observations must lie within the interval \((\mu +- \sigma)\), about 95% of the observations must lie within the interval \((\mu \pm 2\sigma)\), and approximately all the observations must lie within the interval \((\mu \pm 3\sigma)\).
The interval \((\mu \pm \sigma)\) is calculated below:
\((\mu \pm \sigma) = (0.033 — 0.10, 0.033 + 0.10)= (-0.067, 0.133)\).
Note that, the interval \((\mu \pm \sigma)\) for this data set contains negative values, which is not possible according to the given scenario. Evidently, the wider intervals \((\mu \pm 2\sigma)\ and\ (\mu \pm 3\sigma)\) will also contain negative values.
This suggests that the empirical rule does not hold for the given data set.
Hence, the readings on lead content do not have a normal distribution.