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Unit 06 /
Problem 2:


Establishing hypotheses
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Moderate


(Hot Stuff)

Suppose that you're working for the EPA, who has decided that it is safe to drink as much as 5 picocuries per liter. That is, this value of 5 is the dividing line between what is safe, and what is unsafe. (Aside: isn't it wild that there is a bright-line distinction between "safe" and "unsafe"?)

Would you recommend testing $H_{0}: \mu = 5$ vs. $H_{a}: \mu > 5$; or, $H_{0}: \mu = 5$ vs. $H_{a}: \mu < 5$. Explain your reasoning, weighing the consequences of a false-negative, or a false-positive in either case.

Answer:

It depends. See the solution.

Solution:

Setting hypotheses is hard! Even while writing these answers, it can be hard to keep straight what is a false-positive and a false-negative.

If we test the first circumstance, where the alternative hypothesis is $\mu > 5$, then a false-negative would mean that the water is actually unsafe, but our test fails to pick it up; and, a false positive would be that our test incorrectly warns us that the water is unsafe. This circumstance feels like essentially saying, "We'll presume the water is safe, until the test tells us otherwise."

In the second test circumstance, where the alternative hypothesis is that the average is less than five, feels like essentially saying, "We'll presume the water is unsafe, until the test tells us otherwise."

False Negatives

In the first scenario, a false-negative test would have someone drinking bad water; in the second case, a false, negative would have someone drinking bottled water when the water is actually safe.

False Positives

In the first scenario, a false positive would have people drinking bottled water when the water is actually safe; in the second case, a false positive would have someone drinking bad water.

Making a Choice

In order to know which of these alternatives you would like to set up requires a statement about your risk profile, and also the ability of the tests to reject.

A very risk-averse profile would be willing to drink bottled water unless there were very strong evidence that the water was safe. A very risk-accepting profile would be willing to drink tap water unless there was very strong evidence that the water was bad.

  • If the test were perfect, that there were no errors; then there is not difference in these alternatives.
  • If the test were very sensitive, but unspecific (there are a lot of false positives but few false negatives), then different risk profiles might choose one or another alternative.
  • Similarity, if the test were very specific, but not very sensitive (there are few false positives, but a lot of false negatives) then different risk profiles might choose one or another alternative.