Question: You are the manager of a bar that sells beer by the pint (16 oz.) and by the pitcher (64 oz.). The actual amount of beer filled in a pint or pitcher is a random variable, denoted X and Y respectively. You can assume these random variables are independent (the actual amount in the first pint/pitcher is independent from the amount in the next pint/pitcher). The means and standard deviations of X and Y are provided in the table below.
It's Friday and you suspect your bartenders may be habitually giving drinks to their friends for free, which is strictly against the bar's policies. Let M and N be random variables representing the number of pints and pitchers sold, respectively. The means and standard deviations of M and N are provided in the table below. Assume M and N are also independent.
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Statistics of X and Y
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Statistics of M and N
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Size (ounces)
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Mean
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Standard Dev
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Mean
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Standard Dev
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Pint
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16
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14.5
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1.3
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220
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40
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Pitcher
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64
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60.5
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3
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50
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10
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a) Find the expected value and variance of the ounces of beer sold in pits on Friday.
b) Find the expected value and variance of the ounces of beer sold in pitchers on Friday.
c) Find the expected value and variance of the total ounces of beer sold on Friday.
d) Let K be the number of kegs used on Friday. A standard keg holds 15.5 gallons (approximately 1980 oz.). Find the expected value and standard deviation of K. (Hint: K is allowed to take a decimal value. You will need the values from part c. If you couldn't solve part c, use expected value of the total ounces be 6,000 and the variance 700,000).
e) Approximate P(K ≥ 2 7). You may leave your answer in terms of the CDF of the standard normal, denoted Φ(x).
f) The next day, your assistant reports that 2.7 kegs were consumed on Friday, with 200 pints and 40 pitchers accounted for in receipts. Is it time to have a serious conversation with your staff?