Question

Consolidated Power, a large electric power utility, has just built a modern nuclear power plant. This plant discharges waste water that is allowed to flow into the Atlantic Ocean. The Environmental Protection Agency (EPA) has ordered that the waste water may not be excessively warm so that thermal pollution of the marine environment near the plant can be avoided. Because of this order, the waste water is allowed to cool in specially constructed ponds and is then released into the ocean. This cooling system works properly if the mean temperature of waste water discharged is 66^oF or cooler. Consolidated Power is required to monitor the temperature of the waste water. A sample of 100 temperature readings will be obtained each day, and if the sample results cast a substantial amount of doubt on the hypothesis that the cooling system is working properly (the mean temperature of waste water discharged is 66^oF or cooler), then the plant must be shut down and appropriate actions must be taken to correct the problem.

(a) Consolidated Power wishes to set up a hypothesis test so that the power plant will be shut down when the null hypothesis is rejected. Set up the null and alternative hypotheses that should be used.

H₀: µ (Click to select)?> 66 versus H_a: µ (Click to select)?> 66 where µ = mean temperature of waste water.

(b) In the context of this situation, interpret making a Type I error; interpret making a Type II error.

Type I error: decide µ (Click to select)?> 66 (shut down) when µ (Click to select)>? 66 (water is cool enough, no shutdown needed)

Type II error: decide µ (Click to select)>? 66 (do not shut down) when µ (Click to select)>? 66 (water is too warm shutdown needed)

(c) The EPA periodically conducts spot checks to determine whether the waste water being discharged is too warm. Suppose the EPA has the power to impose very severe penalties (for example, very heavy fines) when the waste water is excessively warm. Other things being equal, should Consolidated Power set the probability of a Type I error equal to ? = .01 or ? = .05?

Set ? = 0.05 to make the probability of a (Click to select)Type IIType I error (Click to select)largersmaller.

Answer 1

let the true mean temperature of waste water discharged be

Tthe plant must be shut down and appropriate actions must be taken to correct the problem if the true mean temperature is greater than

tha means Consolidated Power wishes test if the true mean temperature of discharge is greater than

The hypotheses are

b) Type I error is when we reject a null ypothesis that is true. That mean we commit a type I errorwhen we say that th true mean temperatireof discharge is more than 66 F and hence shutdow the plant , when it is actually less than or equal to 66 F.

Type I error decide that (shutdown) when (waer is cool enough, no shutdown required

We comit a type II error when we accept a null hypothess which is false. That is when we say that the true temperature is less than 66 F and no shutdown required, when in fact the temperatire f discharge is more than 66 F

Type II error: Decide (do not shutdown) when (water is too warm sutdown is needed)

c) the EPA has the power to impose very severe penalties (for example, very heavy fines) when the waste water is excessively warm. Tha means Conslidated Power would like to avoid making type II error, that is it would not like accept the null hypothesis when it is false.

Increasing the value of type I error by icreasing the level of significance will reduce the type II error at a given sample size.

That means increasing type I error from 0.01 to 0.05 will make the probability f type II error smaller

Other things being equal, should Consolidated Power set the probability of a Type I error equal to 0.05 to make the probability of a type II error smaller