Find the Value of a Normal Random Variable
-Suppose a brewery has a filling machine that fills 12-ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.15 ounces and a standard deviation of 0.04 ounce. The company is interested in reducing the amount of extra beer that is poured into the 12 ounce bottles. The company is seeking to identify the highest 1.5% of the fill amounts poured by this machine. For what fill amount are they searching
A) 12.237 oz
B) 11.913 oz
C) 12.063 oz
D) 12.087 oz

Question

Quizplus · Accepted Answer

To find the fill amount corresponding to the highest 1.5% of the distribution, we look for the z-score associated with the top 1.5%. This z-score corresponds to the point in the distribution where 98.5% of the values lie below it (since 100% - 1.5% = 98.5%). Looking up the z-score for 98.5% in a standard normal distribution table gives a value of approximately 2.17. Using the formula for converting a z-score to an actual value in a normal distribution, $X = \mu + Z\sigma$, where $\mu = 12.15$ ounces is the mean, $\sigma = 0.04$ ounce is the standard deviation, and $Z = 2.17$, we find $X = 12.15 + 2.17(0.04) = 12.15 + 0.0868 = 12.2368$ ounces, which rounds to 12.237 ounces.