Water specimens 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 discharge is at most 150F, there will be no negative effects on the river's ecosystem. To investigage whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above 150F, researchers will take 50 specimens at randomly selected times and record the temperature of each specimen. The resulting data will be used to test the hypotheses: H0:μ≤150FH0:μ≤150F Ha:μ>150FHa:μ>150F. (a) In the context of this problem, describe Type I and Type II errors. Type I Error: Select an answer A Type I error is not obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is greater than 150F. A Type I error is obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is at most 150F. Type II Error: Select an answer A Type II error is obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is at most 150F. A Type II error is not obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is greater than 150F. (b) Which type of error would you consider more serious? Explain. Select an answer A Type I Error is more serious, as the ecosystem will be harmed and no action will be taken. A Type II Error is more serious, as the ecosystem will be harmed and not action will be taken.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
H0:μ≤150FH0:μ≤150F
Ha:μ>150FHa:μ>150F.
(a) In the context of this problem, describe Type I and Type II errors.
Type I Error:
Select an answer
A Type I error is not obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is greater than 150F.
A Type I error is obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is at most 150F.
Type II Error:
Select an answer
A Type II error is obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is at most 150F.
A Type II error is not obtaining convincing evidence that the mean water temperature is greater than 150F when in fact it is greater than 150F.
(b) Which type of error would you consider more serious? Explain.
Select an answer
A Type I Error is more serious, as the ecosystem will be harmed and no action will be taken.
A Type II Error is more serious, as the ecosystem will be harmed and not action will be taken.
Which of these statements is true about continuous random variables and their probability distributions?
- The distribution is modeled by a probability distribution that is non-negative.
- The probability of an individual
event E is calculated by P(X=E). - The total area under the curve is 1.
- II only
- I and III only
- III only
- I and II only
- I only
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