Concept explainers
The amount of fill in a half-liter (500 ml) soft drink bottle is
a.
Find the mean, so that the probability that a half-liter bottle will not be underfilled is 95%.
Answer to Problem 92CE
Themean, so that the probability that a half-liter bottle will not be underfilled is 95%is 508.225 ml.
Explanation of Solution
Calculation:
The filling of a half-liter or 500 ml bottle of soft drink has a normal distribution with an adjustable mean and a standard deviation of 5 ml.
Normal distribution:
A continuous random variable X is said to follow normal distribution if the probability density function of X is,
Standard normal distribution:
A continuous random variable Z is said to follow standard normal distribution if the probability density function of X is,
The z-score:
The z-score for a variable X with mean
Denote X as the amount of fill in the half-liter bottle. Here,
There must be a 95% chance that the half-liter bottle will not be underfilled. This implies that
Now,
Variable value:
Software procedure:
Step-by-step software procedure to obtain variable value using EXCEL is as follows:
- • Open an EXCEL file.
- • In cell G1, enter the formula “=NORM.S.INV(0.05)”.
- Output using EXCEL software is given below:
Thus,
Hence, themean, so that the probability that a half-liter bottle will not be underfilled is 95% is 508.225 ml.
b.
Find the mean, so that the probability that a half-liter bottle will not be underfilled is 99%.
Answer to Problem 92CE
Themean, so that the probability that a half-liter bottle will not be underfilled is 99% is 511.63 ml.
Explanation of Solution
Calculation:
There must be a 99% chance that the half-liter bottle will not be underfilled. This implies that
Now,
Variable value:
Software procedure:
Step-by-step software procedure to obtain variable value using EXCEL is as follows:
- • Open an EXCEL file.
- • In cell G1, enter the formula “=NORM.S.INV(0.01)”.
- Output using EXCEL software is given below:
Thus,
Thus, the mean, so that the probability that a half-liter bottle will not be underfilled is 99% is 511.63 ml.
c.
Find the mean, so that the probability that a half-liter bottle will not be underfilled is 99.9%.
Answer to Problem 92CE
Themean, so that the probability that a half-liter bottle will not be underfilled is 99.9% is 515.45 ml.
Explanation of Solution
Calculation:
There must be a 99.9% chance that the half-liter bottle will not be underfilled. This implies that
Now,
Variable value:
Software procedure:
Step-by-step software procedure to obtain variable value using EXCEL is as follows:
- • Open an EXCEL file.
- • In cell G1, enter the formula “=NORM.S.INV(0.001)”.
- Output using EXCEL software is given below:
Thus,
Thus, the mean, so that the probability that a half-liter bottle will not be underfilled is 99.9%is 515.45 ml.
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Chapter 7 Solutions
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