A machine fills 64 ounce detergent jugs. It is known that the net amount of detergent poured into each jug has a normal distribution with a standard deviation of 0.35 ounces. The inspector wants to adjust the machine such that at least 95% of the jugs have more than 64 ounces of detergent. what should the mean amount of detergent poured by this machine into these jugs be?
The mean amount of detergent poured by the machine should be 64.35 ounces.
To determine the mean amount of detergent poured by the machine, we can use the Z-score formula. The Z-score is used to standardize values in a normal distribution.
Let's assume that X is the amount of detergent poured by the machine into each jug. We want to find the mean μ, which is the average amount of detergent poured by the machine.
To find the Z-score, we use the formula:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the observed value (64 ounces),
μ is the mean,
σ is the standard deviation (0.35 ounces).
Since we want at least 95% of the jugs to have more than 64 ounces of detergent, we need to find the Z-score corresponding to the 95th percentile.
Using a Z-score table or a statistical calculator, we find that the Z-score corresponding to the 95th percentile is approximately 1.645.
So, we have:
1.645 = (64 - μ) / 0.35
Now, we can solve for μ:
(64 - μ) / 0.35 = 1.645
64 - μ = 1.645 * 0.35
64 - μ = 0.57475
μ = 64 - 0.57475
μ ≈ 63.42525
Therefore, the mean amount of detergent poured by the machine into the jugs should be approximately 63.42525 ounces to ensure that at least 95% of the jugs have more than 64 ounces of detergent.
To find the mean amount of detergent that should be poured into the jugs, we need to determine the cutoff value for the 95th percentile of the normal distribution.
Step 1: Find the Z-score for the 95th percentile.
Since the normal distribution is symmetric, we can find the Z-score using a standard normal distribution table or calculator. The 95th percentile corresponds to a Z-score of approximately 1.645.
Step 2: Use the formula for Z-score to find the mean.
The formula for Z-score is given by:
Z = (x - μ) / σ
Where:
Z is the Z-score,
x is the value of interest,
μ is the mean,
σ is the standard deviation.
In this case, we want to find the mean (μ) and the value of interest (x) is 64 ounces. The Z-score is 1.645 and the standard deviation (σ) is 0.35 ounces.
So, we have:
1.645 = (64 - μ) / 0.35
Step 3: Solve for the mean.
Multiply both sides of the equation by 0.35:
0.35 * 1.645 = 64 - μ
0.57775 = 64 - μ
Rearrange the equation to solve for μ:
μ = 64 - 0.57775
μ ≈ 63.42
Therefore, the mean amount of detergent poured by this machine into these jugs should be approximately 63.42 ounces in order to ensure that at least 95% of the jugs have more than 64 ounces of detergent.