What is the standard normal quantile z* for a 97.5% two-sided interval?

What is the standard normal quantile z* for a 97.5% two-sided interval?

• A. Approximately 2.24
• B. 1.96
• C. 2.58
• D. 3.00

Answer: (A)

The question is about finding the standard normal quantile that gives a 97.5% two-sided (central) interval. For a symmetric, two-sided interval, you want the central area to be 97.5%, leaving the remaining probability split across the two tails. That means each tail has 1.25% beyond the cutoff, so the cutoff z* is the value where the standard normal cumulative distribution equals 0.9875 (since 0.5 plus the 0.975 central proportion gives 0.9875). The value that gives CDF ≈ 0.9875 is about 2.24. So the standard normal quantile for this interval is roughly 2.24, which matches the option around 2.24. Why the other numbers don’t fit: 1.96 corresponds to a 95% central interval (tails of 2.5% each). 2.58 corresponds to a central interval of about 99% (very small tails). 3.00 corresponds to an even wider, about 99.7% central interval.

The question is about finding the standard normal quantile that gives a 97.5% two-sided (central) interval. For a symmetric, two-sided interval, you want the central area to be 97.5%, leaving the remaining probability split across the two tails. That means each tail has 1.25% beyond the cutoff, so the cutoff z* is the value where the standard normal cumulative distribution equals 0.9875 (since 0.5 plus the 0.975 central proportion gives 0.9875).

The value that gives CDF ≈ 0.9875 is about 2.24. So the standard normal quantile for this interval is roughly 2.24, which matches the option around 2.24.

Why the other numbers don’t fit: 1.96 corresponds to a 95% central interval (tails of 2.5% each). 2.58 corresponds to a central interval of about 99% (very small tails). 3.00 corresponds to an even wider, about 99.7% central interval.