studies indicate that drinking water supplied by some old lead-lined city piping systems may contain harmful levels of lead. an important study of the Boston water supply system showed that the distribution of lead content readings for individual water specimens had a mean and standard deviation of approximately .033 milligrams per liter (mg/l) and .10mg/l,respectively.
(a)explain why you believe this distribution is or is not normally distributed.
(b) because the researchers were concerned about the shape of the distribution in part a, they calculated the average daily lead levels at 40 different locations on each of 23 randomly selected days. what can you say about the shape of the distribution of the average daily lead levels from which the sample of 23 days was taken?
(c) what are the mean and standard deviation of the distribution of average lead levels in part b?
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(a) To determine if the distribution of lead content readings in the Boston water supply system is normally distributed, we need to consider a few factors.
Firstly, in a normally distributed data set, the mean, median, and mode are all equal, and they occur at the center of the distribution. In this case, the mean is reported to be approximately 0.033 mg/l.
Secondly, the shape of a normal distribution is symmetric, meaning the data points are evenly distributed on both sides of the mean.
Lastly, the standard deviation measures the spread or dispersion of the data. In this case, the standard deviation is reported to be approximately 0.10 mg/l.
Given these factors, it is reasonable to assume that the distribution of lead content readings in the Boston water supply system is not perfectly normal. The mean and standard deviation may provide some insight into the general shape of the distribution, but further analysis would be required to assess its actual distribution.
(b) With the aim of understanding the shape of the distribution in part (a) and having concerns about it, the researchers calculated the average daily lead levels at 40 different locations on each of the 23 randomly selected days.
When multiple samples are taken from a population, the distribution of the sample means tends to approximate a normal distribution, regardless of the shape of the population distribution. This result is known as the Central Limit Theorem. Therefore, we can expect the distribution of the average daily lead levels from the sample of 23 days to be approximately normal, even if the distribution of individual lead content readings is not.
(c) To determine the mean and standard deviation of the distribution of average lead levels in part (b), we need the data for the average daily lead levels calculated at 40 different locations on each of the 23 randomly selected days. Unfortunately, this information is not provided in the question. Without the actual data, we cannot determine the exact mean and standard deviation.