Suppose a normal distribution has mean $\mu$ and standard deviation $\sigma$. According to the empirical rule, what is the approximate probability that a randomly selected value from this distribution falls within one standard deviation of the mean (that is, between $\mu -\sigma$ and $\mu +\sigma$)?

• A. $68\%$
• B. $50\%$
• C. $95\%$
• D. $99.7\%$

Answer: (A)

Verified step by step guidance

1. Recall the empirical rule (also known as the 68-95-99.7 rule) which describes the approximate percentage of data within 1, 2, and 3 standard deviations from the mean in a normal distribution.
2. Identify that the problem asks for the probability that a value falls within one standard deviation of the mean, i.e., between \(\mu - \sigma\) and \(\mu + \sigma\).
3. According to the empirical rule, approximately 68% of the data in a normal distribution lies within one standard deviation of the mean.
4. This means the probability that a randomly selected value falls between \(\mu - \sigma\) and \(\mu + \sigma\) is about 68%.
5. Therefore, the answer corresponds to the option that states 68%.