In a normal distribution, approximately what percentage of data falls below a value that is $3-\sigma$ below the mean $\mu$?

• A. $2.5\%$
• B. $5\%$
• C. $0.13\%$
• D. $16\%$

Answer: (C)

Verified step by step guidance

1. Recall that in a normal distribution, the mean (\( \mu \)) is the center of the distribution, and the standard deviation (\( \sigma \)) measures the spread of the data around the mean.
2. Understand that the question asks for the percentage of data below a value that is 3 standard deviations below the mean, which corresponds to the point \( \mu - 3\sigma \).
3. Use the properties of the standard normal distribution (mean 0, standard deviation 1) by converting the value \( \mu - 3\sigma \) to a z-score. The z-score here is \( z = -3 \).
4. Consult the standard normal distribution table (z-table) or use a calculator to find the cumulative probability for \( z = -3 \), which gives the proportion of data below this z-score.
5. Express this cumulative probability as a percentage to find the approximate percentage of data falling below 3 standard deviations below the mean.