Help me solve this statistics question.

Question

A drinks machine dispenses coffee into cups. A sign on the machine indicates that each cup contains 50 ml of coffee. The machine actually dispenses a mean amount of 55 ml per cup and 10% of the cup contain less than the amount stated on the sign.



Assuming that the amount of coffee dispensed into each cup is normally distributed,

find the percentage of cups that contain more than 61 ml.

Answer 1

Let X be the random variable measuring the amount of coffee in each cup. We know that X is normally distributed with µ=55 ml and standard deviation σ (unknown for now).

To find the standard deviation, we use the information that P(X<50)=0.1. Looking up a Normal distribution CDF table, we realize that Normal CDF reaches 0.1 at around 1.28σ before the mean (µ). Hence, 50=55-1.28σ or σ=3.9 ml.

The final step is to find P(X>61) (or basically 6ml more than μ). Notice that 6ml is almost 1.54σ. Again looking up the Normal CDF table, we see that CDF reaches 0.938 at μ+1.54σ. This means that 93.8% of cups are below 61ml, or equivalently around 6.2% of cups more than 61ml.