Given a normal distribution with mean $\mu$ = $0$ and standard deviation $\sigma$ = $1$, what are the z-scores that correspond to the boundaries of the middle 95% of the distribution as shown in a standard normal curve graph?

• A. $-2.00$ and $2.00$
• B. $-1.64$ and $1.64$
• C. $-2.58$ and $2.58$
• D. $-1.96$ and $1.96$

Answer: (D)

Verified step by step guidance

1. Understand that the problem asks for the z-scores that mark the boundaries containing the middle 95% of a standard normal distribution, which has mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \).
2. Recall that the middle 95% of the distribution means that 2.5% of the data lies in each tail outside these boundaries, since \( 100\% - 95\% = 5\% \) and this is split equally on both sides.
3. Use the standard normal distribution table (z-table) or a statistical software to find the z-score \( z \) such that the cumulative probability from the left up to \( z \) is 97.5% (which is 95% + 2.5%). This corresponds to the upper boundary.
4. Because the standard normal distribution is symmetric about zero, the lower boundary will be the negative of the upper boundary, i.e., \( -z \).
5. The pair of z-scores \( -z \) and \( z \) you find are the boundaries that contain the middle 95% of the distribution.