Question

The number of gallons of paint that Home Depot sells in a given day is normally distributed with a mean of 197 gallons and a standard deviation of 45 gallons (I realize that the distribution is probably different for weekends compared to weekdays, but just assume everyday has the distribution). How many gallons of paint have to be sold in a given day in order for that day to be in the top 5% of the highest number of gallons sold? (please round your answer to 2 decimal places)

Answer 1

Let X be the number of gallons of paint sold              
X follows normal distribution mean μ and standard deviation σ              
Given    μ = 197 gallons       σ = 45 gallons  
              
Let X' be the number of gallons sold so that the day will be in top 5% for sale              
To find X' such that P(X ≥ X') = 0.05              
that is to find P(X < X') = 1 - 0.05 = 0.95              
From standard normal tables or Excel function NORM.S.INV we find Z' such that              
Z' = NORM.S.INV(0.95)              
Z' = 1.645              
Converting X' to Z-score , Z'              

X' = 45*(1.645) + 197              
X' = 271.03              

Required number of gallons to be sold = 271.03 gallons