Answer:
a) There is a 2.28% probability that a new cup will overflow when filled by the automatic dispenser.
b) The mean amount dispensed by the machine should be set at 16.14 ounces to satisfy this wish.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
Normal model with mean 16 ounces and standard deviation 0.31 ounces. This means that .
A new 16-ounce cup that is being considered for use actually holds 16.62 ounces of drink.
a. What is the probability that a new cup will overflow when filled by the automatic dispenser?
This probability is 1 subtracted by the pvalue of Z when . So
has a pvalue of 0.9772. This means that there is a 1-0.9772 = 0.0228 = 2.28% probability that a new cup will overflow when filled by the automatic dispenser.
b. The company wishes to adjust the dispenser so that the probability that a new cup will overflow is .006. At what value should the mean amount dispensed by the machine be set to satisfy this wish?
This is the value of , with
when Z has a pvalue of 0.94. It is between
and
, so we use
.
The mean amount dispensed by the machine should be set at 16.14 ounces to satisfy this wish.