For the standard normal distribution, what percentage of the area under the curve lies within one standard deviation of the mean (that is, between $\mu -\sigma$ and $\mu +\sigma$)?

• A. About 68%
• B. About 50%
• C. About 95%
• D. About 99.7%

Answer: (A)

Verified step by step guidance

1. Recall that the standard normal distribution is a normal distribution with mean \(\mu = 0\) and standard deviation \(\sigma = 1\).
2. Understand that the question asks for the percentage of the total area under the standard normal curve that lies between \(\mu - \sigma\) and \(\mu + \sigma\), which translates to the interval from \(-1\) to \$1$ in the standard normal distribution.
3. Use the empirical rule (68-95-99.7 rule) which states that approximately 68% of the data in a normal distribution lies within one standard deviation of the mean.
4. Alternatively, you can find this area by calculating the cumulative distribution function (CDF) values at \(z = 1\) and \(z = -1\) and then subtracting: \(P(-1 < Z < 1) = \Phi(1) - \Phi(-1)\), where \(\Phi(z)\) is the CDF of the standard normal distribution.
5. Recognize that this calculation or the empirical rule leads to the conclusion that about 68% of the area under the standard normal curve lies within one standard deviation of the mean.