The Amount of filling a half-liter (500ml)soft drink bottle is normally distributed. The process has a standard deviation of 5ml. The mean is adjustable.
A. Where should the mean be set to ensure a 95% probability that a half-liter bottle will not be under filled?
B. A 99% probability?
C. A 99.9 % probability?
Explain
To answer this question, we need to use the concept of the standard normal distribution and z-scores. A z-score measures the number of standard deviations away from the mean a data point is.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the data value
- μ is the mean
- σ is the standard deviation
In this case, we want to find the mean (μ) that ensures a certain probability that a half-liter bottle will not be under-filled.
A. For a 95% probability:
To find the mean (μ) that ensures a 95% probability, we need to find the z-score that corresponds to this probability. In the standard normal distribution table, a 95% probability corresponds to a z-score of approximately 1.96. Since we are interested in the lower tail of the distribution (to ensure that the bottle is not under-filled), we want to find the z-score that has an area of 0.95 to the left of it. This means we want the z-score that corresponds to a probability of 0.05 in the upper tail. Using the standard normal distribution table, we find the z-score to be approximately 1.96.
Using the formula for z-score, we can now solve for the mean (μ):
1.96 = (500 - μ) / 5
Solving this equation for μ, we get:
500 - μ = 1.96 * 5
500 - μ = 9.8
μ = 500 - 9.8
μ ≈ 490.2
So, the mean should be set to approximately 490.2 ml to ensure a 95% probability that a half-liter bottle will not be under-filled.
B. For a 99% probability:
Similarly, to ensure a 99% probability, we need to find the z-score that corresponds to this probability. In the standard normal distribution table, a 99% probability corresponds to a z-score of approximately 2.58. Using the formula for z-score, we can solve for the mean (μ):
2.58 = (500 - μ) / 5
Solving this equation for μ, we get:
μ = 500 - (2.58 * 5)
μ = 500 - 12.9
μ ≈ 487.1
So, the mean should be set to approximately 487.1 ml to ensure a 99% probability that a half-liter bottle will not be under-filled.
C. For a 99.9% probability:
Again, finding the z-score that corresponds to a 99.9% probability, we get a z-score of approximately 3.29.
Using the formula for z-score, we can solve for the mean (μ):
3.29 = (500 - μ) / 5
Solving this equation for μ, we get:
μ = 500 - (3.29 * 5)
μ = 500 - 16.45
μ ≈ 483.5
So, the mean should be set to approximately 483.5 ml to ensure a 99.9% probability that a half-liter bottle will not be under-filled.