Question

Describing Type I and Type II errors—A power plant discharges water into a river. Regulators determine that as long as the mean temperature of the discharged water is no more than 150°F, there will be no negative effects on the river’s ecosystem. Data will be collected to test the following hypotheses:

?₀: μ = 150°F

?_?: μ > 150°F

A.) Describe a Type I error in the context of this problem

B.) Describe a Type II error in the context of this problem

C.) Which type of error would an environmental group consider the more serious? Explain briefly.

Answer 1

Here as given,

Regulators determine that as long as the mean temperature of the discharged water is less than or equal to 150 F ,there will be no damage to the river ecosystem.

H0: $\mu$ = 150

H1: $\mu$ > 150

1). Type I error is the probability of rejecting H0 when it is actually true . It is denoted by $\alpha$ , that is level of significance.

$\alpha$ = P[ Type I error]

= P[ Rejecting H0 | H0 is true] .

In this context of the problem the type I error will occur when we accept H1 when actually H0 is true , that means when we conclude that the mean temperature of the discharged water is more than 150F when actually it was just equal to 150F . We may wrongly conclude that there will be negative impact on the river ecosystem when actually there is no need to worry about this impact beause the temperature is equal to 150F and actually not exceeding 150F.

2).Type II error is the probability of accepting H0 when actually it is false. It is denoted by $\beta$ .

$\beta$ = P[ Type II error ]

= P[ Accepting H0 | H1 is true].

In this context of the problem the type II error will occur when we accept H0 when actually H1 is true , that means when we conclude that the mean temperature of the discharged water is equal to 150F when actually it is more than 150F . We may wrongly conclude that there will be no negative impact on the river ecosystem when actually there will a serious impact because the temperature of water released will be more than 150F.

3) In the type I error that is when we are concluding that there is a negative impact on the river ecosystem when actually in reality there is no negative impact .This type of wrong conclusion will result into the shut down of the power plant. In type II error we will be in the consideration that there is being no harm to the ecosystem of the river when actually in reality there is a serious harm to the ecosystem. This will result in damaging the river ecosystem. And by the time we know that we concluded wrongly there will be no chance to correct this mistake . And our prime interest lies in discharging the water in the river without harming the ecosystem of that river .

So according to the environmental group the type II error will be more serious.