IM3.7.7 Standard Normal

When calculating probabilities associated with normal distributions, z-scores are used to standardize and compare scores from different distributions.

• A z-score for a particular value measures the number of standard deviations away from the mean.
• A positive z-score corresponds to a value that is above the mean, and a negative z-score corresponds to a value that is below the mean.
• The letter z is used to represent a variable that has a standard normal distribution where the mean is 0 and standard deviation is 1.
• The formula for calculating a z-score for any value x with standard deviation σ and mean μ is .

The U.S. Department of Agriculture (USDA), in its Official Food Plans (www.cnpp.usda.gov), states that the average cost of food for a 14- to 18-year-old male is $261.50 per month. Assume that the monthly food cost for a 14- to 18-year-old male is approximately normally distributed with a mean of $261.50 and a standard deviation of $16.25. The USDA also states that the average cost of food for a 14- to 18-year-old female is $215.20 per month. Assume that the monthly food cost for a 14- to 18-year-old female is approximately normally distributed with a mean of $215.20 and a standard deviation of $14.85. Use z-scores to compare Billy, who spends $270 a month on food, with Gina, who spends $250 a month on food.