How do you convert a z-score to a probability?

How do you convert a z-score to a probability?

• A. Look up Φ(z) in the standard normal CDF to get P(Z ≤ z); for two-tailed tests, use 2Φ(z) − 1.
• B. Use the mean and standard deviation of the data to compute it.
• C. Use P(Z ≥ z) directly from a z-table.
• D. P(Z ≤ z) is always 0.5 regardless of z.

Answer: (A)

Interpreting a z-score as a probability relies on the standard normal cumulative distribution function. If Z is standard normal, P(Z ≤ z) equals Φ(z). So to convert a z-score to a probability, you use Φ(z). In a two-tailed context, the probability that Z falls between −z and z is P(−z ≤ Z ≤ z) = Φ(z) − Φ(−z). By symmetry, Φ(−z) = 1 − Φ(z), so this becomes 2Φ(z) − 1. That’s why the standard approach is to look up Φ(z) and, for two tails, compute 2Φ(z) − 1. The other ideas aren’t correct because you don’t directly compute the probability from the data’s mean and standard deviation without first standardizing to z, and standard tables give Φ(z) = P(Z ≤ z) (with P(Z ≥ z) equal to 1 − Φ(z)). Also, P(Z ≤ z) changes with z, so it isn’t always 0.5.

Interpreting a z-score as a probability relies on the standard normal cumulative distribution function. If Z is standard normal, P(Z ≤ z) equals Φ(z). So to convert a z-score to a probability, you use Φ(z). In a two-tailed context, the probability that Z falls between −z and z is P(−z ≤ Z ≤ z) = Φ(z) − Φ(−z). By symmetry, Φ(−z) = 1 − Φ(z), so this becomes 2Φ(z) − 1. That’s why the standard approach is to look up Φ(z) and, for two tails, compute 2Φ(z) − 1.

The other ideas aren’t correct because you don’t directly compute the probability from the data’s mean and standard deviation without first standardizing to z, and standard tables give Φ(z) = P(Z ≤ z) (with P(Z ≥ z) equal to 1 − Φ(z)). Also, P(Z ≤ z) changes with z, so it isn’t always 0.5.