Explain the 68-95-99.7 rule for the standard normal distribution.

Explain the 68-95-99.7 rule for the standard normal distribution.

• A. About 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ of the mean.
• B. About 68% within ±2σ, 95% within ±3σ, 99.7% within ±4σ.
• C. About 68% within ±0.5σ, 95% within ±1.5σ, 99.7% within ±2.5σ.
• D. Not applicable to the standard normal distribution.

Answer: (A)

The standard normal distribution, with mean 0 and standard deviation 1, follows the empirical rule: about 68% of observations lie within one standard deviation of the mean, about 95% lie within two standard deviations, and about 99.7% lie within three standard deviations. In other words, roughly 68% are between -1 and +1, 95% between -2 and +2, and 99.7% between -3 and +3. This is why the statement describing those intervals around the mean is the best description. The other options propose different ranges that don’t match these percentages, or claim it doesn’t apply to the standard normal, which isn’t correct. More precise values are about 68.27%, 95.45%, and 99.73%, but the rounded rule captures the idea accurately.

The standard normal distribution, with mean 0 and standard deviation 1, follows the empirical rule: about 68% of observations lie within one standard deviation of the mean, about 95% lie within two standard deviations, and about 99.7% lie within three standard deviations. In other words, roughly 68% are between -1 and +1, 95% between -2 and +2, and 99.7% between -3 and +3. This is why the statement describing those intervals around the mean is the best description. The other options propose different ranges that don’t match these percentages, or claim it doesn’t apply to the standard normal, which isn’t correct. More precise values are about 68.27%, 95.45%, and 99.73%, but the rounded rule captures the idea accurately.