Percentage of cartoons with the 68-95-99.7 rule

Question

Milk cartons are filled by a machine. The mean volume of a carton of milk is μ = 1.0 quart and the
standard deviation is σ = 0.06 quart. Assume that the volume of milk in the cartons is normally
distributed.

 

What percent of the cartons will contain at least 0.94 quarts?

 

What percent of the cartons will contain at most 1.12 quarts?

 

 

Answer 1

The empirical rule (68-95-99.7) says that 68% of the data in a normal distribution is within ± 1 standard deviation, 95% is within ± 2 standard deviations, and 99.7% is within ± 3 standard deviations of the mean.

 

Since a normal distribution is symmetrical around the mean, we can use that fact and the empirical rule to answer the questions.

 

What percent of the cartons have at least 0.94 quarts? 

At least 0.94 quarts is anything greater than or equal to the mean - 1 standard deviation. 34% (1/2 of 68) of the data is between the mean and -1 standard deviation. Since the mean and above has 50% of the data, 84% of the cartons will have at least 0.94 quarts

 

What percent has at most 1.12 quarts?

At most 1.12 quarts is anything less than or equal to the mean + 2 standard deviations. We know that 50% is below the mean, 34% is between the mean and + 1 standard deviation, and 13.5% is between 1 and 2 standard deviations above the mean.  The total that has at most 1.12 quarts is 50+34+22.5 = 97.5%