More on the Choice of Alternate Hypotheses

Here's a longer version of my explanation of that first clicker question today, for those interested:

Clicker Question: Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150 degrees Fahrenheit, there will be no negative effects on the river's ecosystem. To investigate whether the plan is in compliance with the regulations that prohibit a mean discharge water temperature above 150 degrees, 50 water samples will be taken at randomly selected times, and the temperatures of each sample recorded. Which of the following hypothesis tests should be used?

  1. H0: μ = 150 vs. HA: μ < 150
  2. H0: μ = 150 vs. HA: μ > 150

If we were to get a low p-value (say, p=0.02) with option 1, then that would be strong evidence that the water temperature is less than 150 degrees. (There would be only a 2% chance that we would get water temperatures as low as, say, 145 degrees if the temperature were really greater than 150. Since this probability is so low, we would conclude that the water temperature must be lower than 150.)

If we were to get a low p-value (say, p=0.02) with option 2, then that would be strong evidence that the water temperature is greater than 150 degrees. (There would only be a 2% chance that we would get water temperatures as big as, say, 155 degrees if the temperature were really less than 150. Since this probability is so low, we would conclude that the water temperature must be greater than 150.)

Now suppose you're the power plant owner, and you want to avoid an unnecessary fine by the EPA for high discharge water temperatures. You would want to use option 2 since you'll want to see strong evidence that your water temperature is too high. If you got a low p-value, then that would be strong evidence that your water temperature is too high, so you would go along with the EPA fine. If you got a high p-value, then that wouldn't be strong evidence that your water temperature is too high, so the EPA presumably wouldn't fine you. A low p-value minimizes the chance of an unnecessary fine.

Now suppose you're Green Peace and you really don't want high water temperatures to kill fish. You would want to use option 1 since you'll want to see strong evidence that the water temperature is low enough. If you got a low p-value, then that would be strong evidence that the water temperature is low enough, so you would be okay with the EPA not fining the power plant. If you got a high p-value, then there would not be strong evidence that the water temperature is low enough, so you would want the EPA to go ahead and fine the power plant. A low p-value minimizes the chance that the power plant will "get away" with too-high water temperatures.

In practice, you would typically collect your water samples and compute the sample mean. If the mean were bigger than 150, then if you're the power plant, then you would want to conduct an option-2 hypothesis test to see if there's really sufficient evidence that your water is too hot. If the mean were bigger than 150 and you're Green Peace, then you would just say, ``Fine the power plant.''

If the mean were lower than 150, then if you're the power plant, then you would tell the EPA not to fine you since your water is cool enough. If the sample mean were lower than 150 and you're Green Peace, then you would want to conduct an option-1 hypothesis test to see if there's really sufficient evidence that the water is cool enough.

Image: "Backwash 5," Pulpolux, Flickr (CC)

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