A machine for filling 2-liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation 0.002 liter and mean whatever amount the machine is set to deliver.
If the machine is set to deliver 2 liters (so the mean amount delivered is 2 liters) what proportion of the bottles will contain at least 2 liters of soft drink?
Find the minimum setting of the mean amount delivered by the machine so that at least 99% of all bottles will contain at least 2 liters.
50%
Look at Z score table in the back of the statistics text for 99% probability and its Z score. Insert Z score into equation below.
Z = (score-mean)/SD
To find the proportion of bottles that will contain at least 2 liters, we need to calculate the area under the normal distribution curve to the right of 2 liters. This corresponds to finding the probability that the amount delivered is greater than or equal to 2 liters.
For a normally distributed variable with a given mean (μ) and standard deviation (σ), we can use the z-score formula to standardize the variable. The z-score formula is given by:
z = (x - μ) / σ
where x is the value of interest (2 liters in this case), μ is the mean, and σ is the standard deviation.
We can then use a standard normal distribution table or calculator to find the proportion of values that are greater than or equal to the z-score corresponding to 2 liters.
Using the given standard deviation of 0.002 liters and a mean of 2 liters, the z-score can be calculated as follows:
z = (2 - 2) / 0.002
z = 0
Since the z-score is 0, the proportion of bottles that will contain at least 2 liters is equal to 0.5 or 50%.
To find the minimum setting of the mean amount delivered by the machine so that at least 99% of all bottles will contain at least 2 liters, we need to find the corresponding z-score that corresponds to the upper 1% of the distribution.
Using a standard normal distribution table or calculator, we can find the z-score that corresponds to the upper 1% (0.01). The z-score is approximately equal to 2.33.
Using the z-score formula, we can calculate the minimum mean (μ) as follows:
z = (2 - μ) / 0.002
2.33 = (2 - μ) / 0.002
Solving for μ, we get:
2 - μ = 2.33 * 0.002
2 - μ = 0.00466
μ = 2 - 0.00466
μ = 1.99534
Therefore, the minimum setting of the mean amount delivered by the machine should be approximately 1.99534 liters to ensure that at least 99% of all bottles will contain at least 2 liters.
To find the proportion of bottles that will contain at least 2 liters of soft drink, we need to use the concept of the standard normal distribution.
For the first part of the question, the mean amount delivered by the machine is set to 2 liters, and the standard deviation is 0.002 liters. We want to find the proportion of bottles that will be filled with at least 2 liters.
Step 1: Convert the problem to a standard normal distribution.
To convert the problem to a standard normal distribution, we need to find the z-score (also known as standardized value) corresponding to the value of 2 liters. The z-score is calculated by subtracting the mean and dividing by the standard deviation.
z = (x - μ) / σ
In this case, x (the value we want to convert) is 2 liters, μ (mean) is 2 liters, and σ (standard deviation) is 0.002 liters.
z = (2 - 2) / 0.002
z = 0
Step 2: Find the proportion from the standard normal distribution table.
Using the standard normal distribution table or calculator, we can find the proportion (or cumulative probability) associated with the z-score of 0. This proportion represents the area under the standard normal curve to the left of the z-score.
From the standard normal distribution table, we find that the proportion corresponding to a z-score of 0 is 0.5.
Therefore, 0.5 or 50% of the bottles will contain at least 2 liters of soft drink.
For the second part of the question, we need to find the minimum setting of the mean amount delivered by the machine so that at least 99% of all bottles will contain at least 2 liters.
Step 1: Convert the problem to a standard normal distribution.
We know that we want to find the minimum mean such that at least 99% of the bottles will contain at least 2 liters. We need to find the z-score corresponding to the proportion 99%.
Step 2: Find the z-score from the standard normal distribution table.
From the standard normal distribution table, we find the z-score corresponding to a proportion of 99% or 0.99. The z-score is approximately 2.33.
Step 3: Convert the z-score to the mean value.
Using the formula z = (x - μ) / σ, we need to rearrange the formula to solve for μ, the mean.
μ = x - z * σ
In this case, x is 2 liters, z is 2.33, and σ is 0.002 liters.
μ = 2 - 2.33 * 0.002
μ ≈ 1.99534
Therefore, the minimum setting of the mean amount delivered by the machine should be approximately 1.99534 liters to ensure that at least 99% of all bottles will contain at least 2 liters of soft drink.