Consider the following test of hypothesis:

Certain types of mining operations release mildly radioactive byproducts. These byproducts may be released into the freshwater supply during the processing of ores. The EPA has issued a regulation that sets the maximum level for naturally occurring radiation in drinking water at 5 picocuries per liter. To see if a city’s water supply is safe, a random sample of 16 water specimens is selected, and the radiation in each specimen is measured in picocuries per liter. These data are below.

Using a = .01, is there sufficient evidence to indicate the mean level of radiation is safe (below the EPA’s maximum level)?

The parameter of interest is m, the mean level of radiation in a city’s water supply, and the hypotheses to be tested are:

                H₀: m = 5 (The water is not safe.)

                H_a: m < 5 ( The water is safe.)

D.  Discuss the consequences of making a Type II error.

E.  Considering your answers above, would you want a , b , or both near zero? Carefully explain why.

A.  A Type I error would be committed if it is concluded the water is safe when in reality the water is contaminated.

B.  A Type II error would be committed if it is concluded the water is not safe when in fact the water is safe.

C.  If it is erroneously concluded that the water is safe (when in fact it is not safe) people will continue to drink radioactively contaminated water, which is very dangerous and thus a very serious error.

D.  If it is concluded the water is not safe (when it is safe) people would be alerted and they would not use the water. More tests of the water would be conducted, and hopefully it would soon be discovered that the water really is safe. This would be an inconvenience for those using the water, but not life threatening. This error is not as serious as a Type I error.

E.  Since a is how frequently a Type I error is made, and a Type I error could cause serious illness or death, the value of a should be as close to zero as possible. Since a Type II error is not as serious, and b is how frequently this error is made, the value of b does not have to be as close to zero. We would still want the value of b to be fairly small so that the water supply is not needlessly shut off very frequently.

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