Which of the following properties can numbers have?
1 answer:
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Answer:
(A) Type I error in the context of this problem is chances of regulators believing that the mean temperature of the discharged water is more than 150°F but in actual the mean temperature of the discharged water was 150°F.
(B) Type II error in the context of this problem is chances of regulators believing that the mean temperature of the discharged water is no more than 150°F but in actual the mean temperature of the discharged water was more than 150°F.
(C) An environmental group will consider the Type II error more serious.
Step-by-step explanation:
We are given that 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.
We are also given with the following hypothesis;
Null Hypothesis, :
= 150°F
Alternate Hypothesis, :
> 150°F
(A) <u><em>Type I error</em></u><em> states that Probability of rejecting null hypothesis given the fact that null hypothesis was true or in other words Probability of rejecting a true hypothesis.</em>
So, Type I error in the context of this problem is chances of regulators believing that the mean temperature of the discharged water is more than 150°F but in actual the mean temperature of the discharged water was 150°F.
(B) <u><em>Type II error </em></u><em>states that Probability of accepting null hypothesis given the fact that null hypothesis was false or in other words Probability of accepting a false hypothesis.</em>
So, Type II error in the context of this problem is chances of regulators believing that the mean temperature of the discharged water is no more than 150°F but in actual the mean temperature of the discharged water was more than 150°F.
(C) An environmental group will consider the Type II error more serious because by committing Type II error they believe that the mean temperature of the discharged water is no more than 150°F and they assume that there is no negative effects on the river’s ecosystem but in reality that the mean temperature of the discharged water was more than 150°F and it is producing negative effects on the river’s ecosystem.
Answer:
Remember, if B is a set, R is a relation in B and a is related with b (aRb or (a,b))
1. R is reflexive if for each element a∈B, aRa.
2. R is symmetric if satisfies that if aRb then bRa.
3. R is transitive if satisfies that if aRb and bRc then aRc.
Then, our set B is .
a) We need to find a relation R reflexive and transitive that contain the relation
Then, we need:
1. That 1R1, 2R2, 3R3, 4R4 to the relation be reflexive and,
2. Observe that
- 1R4 and 4R1, then 1 must be related with itself.
- 4R1 and 1R4, then 4 must be related with itself.
- 4R1 and 1R2, then 4 must be related with 2.
Therefore is the smallest relation containing the relation R1.
b) We need a new relation symmetric and transitive, then
- since 1R2, then 2 must be related with 1.
- since 1R4, 4 must be related with 1.
and the analysis for be transitive is the same that we did in a).
Observe that
- 1R2 and 2R1, then 1 must be related with itself.
- 4R1 and 1R4, then 4 must be related with itself.
- 2R1 and 1R4, then 2 must be related with 4.
- 4R1 and 1R2, then 4 must be related with 2.
- 2R4 and 4R2, then 2 must be related with itself
Therefore, the smallest relation containing R1 that is symmetric and transitive is
c) We need a new relation reflexive, symmetric and transitive containing R1.
For be reflexive
- 1 must be related with 1,
- 2 must be related with 2,
- 3 must be related with 3,
- 4 must be related with 4
For be symmetric
- since 1R2, 2 must be related with 1,
- since 1R4, 4 must be related with 1.
For be transitive
- Since 4R1 and 1R2, 4 must be related with 2,
- since 2R1 and 1R4, 2 must be related with 4.
Then, the smallest relation reflexive, symmetric and transitive containing R1 is