a machine fills containers with a standard deviation of filling weights of 1.5. calculate the mean weight if only 4 % of the containers hold less than 19 ounces
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion and its Z score. Insert the Z score and the other values in the above equation and solve for the mean.
To calculate the mean weight of the containers, we need to use the concept of a normal distribution and the cumulative distribution function.
Step 1: Find the z-score
To determine the z-score, which represents the number of standard deviations away from the mean, we use the formula:
z = (X - μ) / σ
where X is the observed value, μ is the mean, and σ is the standard deviation.
In this case, we have X = 19 ounces and σ = 1.5 (standard deviation). Let's find the corresponding z-score.
z = (19 - μ) / 1.5
Step 2: Find the cumulative distribution function (CDF)
The CDF represents the probability of a variable being less than or equal to a given value. Since we want to find the mean weight when only 4% of the containers hold less than 19 ounces, we need to find the CDF value for that z-score.
Using a standard normal distribution table or a calculator with a normal distribution function, we can find the z-score (let's call it z1) that corresponds to a cumulative probability of 4%. This means we need to find the value where P(Z ≤ z1) = 0.04.
Step 3: Solve for the mean (μ)
Now that we have z1, the z-score corresponding to a cumulative probability of 4%, we can substitute it into the z-score formula from Step 1 and solve for μ.
z1 = (19 - μ) / 1.5
Solving this equation for μ will give us the mean weight of the containers.
Please note that you may need to use a calculator, spreadsheet, or statistical software to perform these calculations accurately.