If an observation has a z-score of $0$ in the standard normal distribution, what does this indicate about the observation's value relative to the mean?

• A. The observation is at the maximum value of the distribution.
• B. The observation is one standard deviation above the mean.
• C. The observation is one standard deviation below the mean.
• D. The observation is exactly equal to the mean.

Answer: (D)

Verified step by step guidance

1. Recall that a z-score represents the number of standard deviations an observation is from the mean in a standard normal distribution.
2. Understand that the formula for a z-score is given by: \(z = \frac{X - \mu}{\sigma}\), where \(X\) is the observation, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
3. If the z-score is 0, substitute \(z = 0\) into the formula: \$0 = \frac{X - \mu}{\sigma}$.
4. Multiply both sides of the equation by \(\sigma\) to isolate \(X - \mu\): \$0 \times \sigma = X - \mu\( which simplifies to \)0 = X - \mu$.
5. Solve for \(X\) to find that \(X = \mu\), meaning the observation is exactly equal to the mean.