Fifteen (15) samples of water were collected from a certain treatment facility in order to gain some insight regarding the amount of coliform in the urban pipeline. The concentration of the coliform is measured in parts per million (ppm), per liter. Suppose that the mean at the treatment facility is not as important as the upper extreme of the distribution of the amount of coliform detected. The concern is whether the concentration of coliform is too large. Readings for the 15 water samples gave a sample mean of 3.84 ppm per liter and a standard deviation of 3.07 ppm per liter. Assume that the readings are a random sample from a normal distribution. Calculate a Prediction Interval (upper 95% prediction limit) and a Tolerance Limit (95% upper tolerance limit that exceeds 95% of the population values). Interpret both; that is, tell what each communicates about the upper extreme of the distribution of coliform at the sampling in the treatment facility. a. Prediction Upper Limit: Xn+1 ≤ 9.424. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 11.72. Hence, we are 95% confident that a limit of 11.72 will contain 95% of the coliform measurement in the water samples b. Prediction Upper Limit: Xn+1 ≤ 9.398. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 11.86. Hence, we are 95% confident that a limit of 11.86 will contain 95% of the coliform measurement in the water samples c. Prediction Upper Limit: Xn+1 ≤ 10.597. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 11.72. Hence, we are 95% confident that a limit of 11.72 will contain 95% of the coliform measurement in the water samples d. Prediction Upper Limit: Xn+1 ≤ 10.641. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 12.91. Hence, we are 95% confident that a limit of 12.91 will contain 95% of the coliform measurement in the water samples
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.
Fifteen (15) samples of water were collected from a certain treatment facility in order to gain some insight regarding the amount of coliform in the urban pipeline. The concentration of the coliform is measured in parts per million (ppm), per liter. Suppose that the
Calculate a Prediction Interval (upper 95% prediction limit) and a Tolerance Limit (95% upper tolerance limit that exceeds 95% of the population values). Interpret both; that is, tell what each communicates about the upper extreme of the distribution of coliform at the sampling in the treatment facility.
a. |
Prediction Upper Limit: Xn+1 ≤ 9.424. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 11.72. Hence, we are 95% confident that a limit of 11.72 will contain 95% of the coliform measurement in the water samples |
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b. |
Prediction Upper Limit: Xn+1 ≤ 9.398. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 11.86. Hence, we are 95% confident that a limit of 11.86 will contain 95% of the coliform measurement in the water samples |
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c. |
Prediction Upper Limit: Xn+1 ≤ 10.597. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 11.72. Hence, we are 95% confident that a limit of 11.72 will contain 95% of the coliform measurement in the water samples |
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d. |
Prediction Upper Limit: Xn+1 ≤ 10.641. This means that a new observation will have a chance of 95% to fall below the upper limit. Tolerance Upper Limit: 12.91. Hence, we are 95% confident that a limit of 12.91 will contain 95% of the coliform measurement in the water samples |
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