Unit 9: Sampling Distributions

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 Milestone

Detecting Arsenic in Our Drinking Water

Recently the safe level of arsenic in drinking water has been under some debate. We know that the content of drinking water usually contains impurities in such micro-quantities as to be harmless. However, since arsenic levels above a certain level can be poisonous, it is important to monitor arsenic levels at purification plants. In this milestone, we'll use simulations to understand which statistic, the mean or the maximum, will help us better detect an increase in arsenic.

Write up your answers to the questions as well as any summarizing thoughts, and send the file to your instructor. Call the file ms9yourlastname and drop this in the Drop Box.


1. To ensure water safety, quality control engineers take six water samples every hour and compute the mean of their arsenic contents. If the mean exceeds a certain predetermined amount, then the purification process is declared faulty. The engineers know that the arsenic content of water from this purification plant is normally distributed with a mean of 25 micrograms per liter (µg/l) and a standard deviation of 5.1 µg/l.


a) Devise a mathematical simulation that will generate the six water samples from the original water supply and calculate the mean of their arsenic content. Repeat the simulation 100 times and plot the means. Describe the distribution; its shape, center, and spread.

b) Suppose a new arsenic source adds 8 µg/l to all water samples, how will the distribution of arsenic levels in the water change?
Now describe a mathematical simulation that will generate the six water samples from the new water supply that has the higher arsenic level and calculate the mean of their arsenic content. Repeat the simulation 100 times and plot the means.

c) Describe the differences in the two distributions. Was it easy to detect the change in arsenic content?


2. A second group of engineers at the same water treatment plant takes six water samples every hour and computes the maximum of their arsenic content.


a) Devise a mathematical simulation that will generate six water samples from the original water supply and calculate the maximum of their arsenic content. Repeat the simulation 100 times and plot the maximums.

b) Suppose a new arsenic source adds 8 µg/l to all water samples, how will the distribution of arsenic levels in the water change?
Now describe a mathematical simulation that will generate the six water samples from the new water supply that has the higher arsenic level and calculate the maximum of their arsenic content. Repeat the simulation 100 times and plot the maximums.

c) Describe the differences in the two distributions? Was it easy to detect the change in arsenic content?


3. Was it easier to detect the difference when we used the mean or when we used the maximum? Explain your reasoning.


Here are directions for using the TI sequence option in our simulation.

We can generate a sample of six values from a normal distribution with mean = 25 and standard deviation = 5.1 by using MATH PRB 6: randnorm(25, 5.1,6) and then record the mean. Repeating the simulation would be tedious so we could write a program or use the sequence option; sequence (equation, counter, lower, upper)
seq (mean(randnum(25, 5.1,6)), X, 1, 100) STO L1
2nd-List OPS ( 2ndLIST MATH 3:mean (MATH PRB 6: randnorm(25, 5.1, 6)), X, 1, 100) STO L1.
This sequence of commands will store the mean for 100 samples of size = 6 in List 1 so that we can plot these means.