11A In-Class Activity

As we’ve seen, statistics is an important tool that helps us make inferences about a population when we are only able to consider a sample. Hypothesis testing can be used to explore whether there has been a change in a particular population parameter or whether a parameter is actually different than what had previously been assumed.

In 2014, residents of Flint, Michigan began to suspect their water was contaminated with lead after the city switched its water source. The Michigan Department of Environmental Quality (DEQ) conducted an investigation and found Flint’s water to be compliant with federal water safety regulations, which require that under 10% of homes yield water samples that have lead levels of 15 parts per billion (ppb) or above. However, the residents of the city were not convinced, and they took their own sample of residences as part of the Flint Water Study (FWS).[1]

Credit: iStock/kedsanee

Question 1

A water sample is “contaminated” if it contains lead levels of 15 ppb or above. What  percentage of residences in the FWS’s sample returning contaminated water  samples would convince you that the actual proportion of all residences with contaminated water in Flint was above 10%?

Question 2

It was the onus of the FWS to convince the U.S. Environmental Protection Agency (EPA), the governor, and other officials that the city of Flint was actually not  compliant with federal water safety regulations. In other words, it was their  responsibility to provide convincing evidence that the current assumption at that time (that under 10% of residences in the city had contaminated water) was false.

  1. What is the population parameter of interest in this situation?
  2. The compliance threshold for water safety is 10%. Let’s suppose a worst case scenario that Flint met this threshold exactly. Considering this situation  as a hypothesis test, what is the null hypothesis?
  3. The residents who lead the charge in the FWS were convinced there was a  higher proportion of homes in the city with contaminated water than is  allowable under federal guidelines. What was their alternative hypothesis?

Question 3

In the FWS’s sample, describe what would have constituted strong evidence that the city of Flint was not actually compliant with federal water safety guidelines (i.e.,  strong evidence against the null hypothesis that the city was compliant and strong  evidence that there was a higher proportion of homes with lead-contaminated water  than had been believed). Explain.

Question 4

The FWS took a sample of 271 residences in Flint spread throughout the city, and  they found contaminated water samples in 20% of their sample.[2]

  1. What is the sample proportion?
  2. The DEQ claimed that the city was compliant with federal guidelines and only  10% of residences actually had contaminated water. According to the DEQ  (and our corresponding null hypothesis), what was the assumed population  proportion of residences with contaminated water?

How likely would it have been for the FWS to obtain a sample with a proportion of  contaminated residences as high as 0.20 if the city was actually compliant? We can use  the sampling distribution of the sample proportion and the normal distribution to  determine this probability.

Since the normal distribution is continuous, it does not make sense to consider the  probability of obtaining a sample with exactly 20% of the residences contaminated (which would be 0). So, since we’re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other  words, how likely are we to have a sample proportion fall in that high range of 20% or  more if the true population proportion is only 10%?

n this case, if the actual proportion of contaminated residences was only 10%, the probability of obtaining a sample with 20% or more of residences yielding contaminated samples would be:

[latex]P(\hat{p}\geq 0.2) = 0.000002[/latex]

Question 5

What can you conclude from this probability? Would it have been very likely for the FWS to obtain a sample proportion of 0.20 or higher if the city was actually compliant?

Question 6

What are the two possible explanations for the high proportion of residences with  contaminated water that the FWS found?

Question 7

Consider the large proportion of residences with contaminated water found by the  FWS. What would you conclude about the water in Flint based on that sample?

Question 8

In the context of the hypothesis test, would you reject or fail to reject the null  hypothesis?

Question 9

Notice that in conducting this hypothesis test, we began by assuming the null  hypothesis was true (i.e., it was already the assumption that Flint was compliant with  federal regulations). The FWS then collected evidence in support of the alternative— that Flint was not compliant. The possibilities are then that the FWS obtained  enough evidence to reject the null hypothesis or that it did not obtain enough  evidence to reject the null hypothesis. As we saw in the preview assignment, the  only possible conclusions of a hypothesis test are to reject the null hypothesis or to  fail to reject the null hypothesis. There is never a conclusion to a hypothesis test in  which one “accepts” the null hypothesis.

In your own words, describe the only possible conclusions of this hypothesis test,  and explain why it would not be valid to conclude from the probabilistic evidence in this test that Flint was compliant with federal regulations.

Question 10

Based on the results of the FWS, as well as a study done by a pediatrician on the  blood lead levels of children in Flint, the city switched the water supply back to the  previous source, and the governor declared a state of emergency a few months  later.

Summarize how the FWS provided evidence that contributed to these decisions.

 

  1. Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https://fivethirtyeight.com/features/what-went-wrong-in-flint-water-crisis-michigan/ ↵
  2. Barry-Jester, A. M. (2016, January 26). What went wrong in Flint. FiveThirtyEight. https://fivethirtyeight.com/features/what-went-wrong-in-flint-water-crisis-michigan/ ↵