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?
a. The standard deviation is larger than mean. You cannot have negative values of lead in the water.
b. It would probably be similar to the previous data. Were the locations randomly or representatively selected?
c. With the data on your post, I have no idea, unless it corresponds to the previous distribution.
(a) To determine whether the distribution is normally distributed, we can visually inspect the data using a histogram or conduct a statistical test, such as the Shapiro-Wilk test or the Anderson-Darling test.
Histogram: Plotting a histogram of the lead content readings can give us a visual representation of the distribution. If the histogram shows a roughly bell-shaped curve, it suggests a normal distribution. However, if the histogram displays skewness or a non-symmetrical shape, it indicates a deviation from normality.
Statistical test: The Shapiro-Wilk test or Anderson-Darling test can provide a numerical assessment of the distribution's departure from normality. These tests generate a p-value, which measures the probability that the observed distribution deviates significantly from a normal distribution.
(b) The researchers calculated the average daily lead levels at 40 different locations on each of 23 randomly selected days. As per the Central Limit Theorem, when we take large random samples, regardless of the shape of the population distribution, the sampling distribution of the sample mean tends to be more closely approximated by a normal distribution. Hence, we can assume that the shape of the distribution of the average daily lead levels from which the sample of 23 days was taken is approximately normal.
(c) To calculate the mean and standard deviation of the distribution of average lead levels in part b, we need the sample data. Since you haven't provided the actual data, we cannot compute the specific values. However, the mean of the distribution of average lead levels is expected to be relatively close to the mean of the individual lead content readings, which was approximately 0.033 mg/l. The standard deviation of the distribution of average lead levels, often known as the standard error, is obtained by dividing the standard deviation of the individual readings by the square root of the sample size. In this case, since 40 locations were sampled for each of the 23 days, we divide the standard deviation of approximately 0.10 mg/l by the square root of 40 to determine the standard error.