Answer:
a) 
And we can find this probability using the complement rule and the standard normal table or excel:
b)
And we can find this probability using the standard normal table or excel:
c) For this case we have a total of 868 cups and we know that the fraction of cups that will be overflow is 0.159, so then the expected number of cups overflow would be:
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the amount of soft frink of a population, and for this case we know the distribution for X is given by:
Where 
and 
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the complement rule and the standard normal table or excel:
Part b
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the standard normal table or excel:
Part c
For this case we have a total of 868 cups and we know that the fraction of cups that will be overflow is 0.159, so then the expected number of cups overflow would be:
