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Difficulty: MediumNormal Distributions, Standard Deviation Curves, and Percentile Ranks

The annual snowfall totals in a high-altitude meteorological district are normally distributed with a mean of 140140 inches and a standard deviation of 1212 inches. According to the 68–95–99.7 empirical rule for normal distributions, what percent of the years have an annual snowfall between 116116 inches and 152152 inches?

Answer: 81.5 %

Answer

81.5%
To find the percentage of data between 116116 inches and 152152 inches, first calculate the standard deviation distances (zz-scores) from the mean of 140140 inches. 116116 inches is 2424 inches below the mean, which corresponds to z=−2z = -2. 152152 inches is 1212 inches above the mean, which corresponds to z=+1z = +1. According to the 68–95–99.7 empirical rule, 95%95\% of the data lies within 22 standard deviations of the mean, meaning 47.5%47.5\% lies between z=−2z = -2 and z=0z = 0. Similarly, 68%68\% of the data lies within 11 standard deviation of the mean, meaning 34%34\% lies between z=0z = 0 and z=+1z = +1. Summing these two symmetric halves gives 47.5%+34%=81.5%47.5\% + 34\% = 81.5\%.

Step-by-Step Solution

1
Convert the boundary values (116116 inches and 152152 inches) into standard z-scores.
zlower=116−14012=−2z_{lower} = \frac{116 - 140}{12} = -2 and zupper=152−14012=+1z_{upper} = \frac{152 - 140}{12} = +1
Standardizing raw values into z-scores allows the application of standard normal distribution properties.
2
Apply the 68–95–99.7 empirical rule to split the area relative to the mean (z=0z = 0).
Area from z=−2z = -2 to z=0z = 0 is 47.5%47.5\%; Area from z=0z = 0 to z=+1z = +1 is 34%34\%.
The normal curve is symmetrical around the mean. Thus, 95%95\% between −2σ-2\sigma and +2σ+2\sigma yields 47.5%47.5\% below the mean, and 68%68\% between −1σ-1\sigma and +1σ+1\sigma yields 34%34\% above the mean.
3
Sum the percentages of the two disjoint regions bounded by z=−2z = -2 and z=+1z = +1.
47.5%+34%=81.5%47.5\% + 34\% = 81.5\%
Combining the area below the mean and the area above the mean gives the total percentage of values falling within the specified interval.

Key Concept

Normal distribution empirical rule (68–95–99.7 rule) with asymmetric standard deviation boundaries
Estimated Time:1m 30s
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Topics

Normal Distributions, Standard Deviation Curves, and Percentile Ranks

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