Empirical Rule Worked Example

Q6. The vodka dispensing machine keeps malfunctioning. However----it can be regulated to give µ ounces (I don`t know: what measures did they use?). If the ounces of fill are normally distributed with a standard deviation of 0.4 ounces, at what value should we set µ so that a 6 ounce vodka mug will overflow only 2.5% of the time?

Answer: We know from the Empirical Rule that 95% of the observations are within 2 standard deviations of the mean. So that means that 5% of the observations are in the tails to the left and right of 95%. The normal distribution is symmetric, so each of the two tails contains 2.5% of the observations. We want to set the machine so that 6 ounces marks the point where 2.5% of the drinks overflow (dreadful waste of vodka if you ask me!). Look at my beautiful sketch. Make sure you can understand why having 2.5% in one tail means that the random variable that corresponds to that area must be 2 standard deviations from the mean.

We’re given that the standard deviation is 0.4 ounces. Two standard deviations is 0.4 * 2 = 0.8. So µ must be 0.8 ounces from 6 ounces. So we should set µ = 6 – 0.8 = 5.2 ounces.