Using the standard normal distribution, what is the value of $P(0<z<2.25)$?

• A. $0.4432$
• B. $0.4878$
• C. $0.3944$
• D. $0.5122$

Answer: (B)

Verified step by step guidance

1. Understand that the problem asks for the probability that the standard normal variable \( z \) lies between 0 and 2.25, i.e., \( P(0 < z < 2.25) \).
2. Recall that the standard normal distribution is symmetric about zero, and the total area under the curve is 1. The mean is 0 and the standard deviation is 1.
3. Use the cumulative distribution function (CDF) of the standard normal distribution, denoted as \( \Phi(z) \), which gives \( P(Z \leq z) \). To find \( P(0 < z < 2.25) \), calculate \( \Phi(2.25) - \Phi(0) \).
4. Look up the values of \( \Phi(2.25) \) and \( \Phi(0) \) in the standard normal table or use a calculator with the standard normal CDF function. Note that \( \Phi(0) = 0.5 \) because 0 is the mean.
5. Subtract \( \Phi(0) \) from \( \Phi(2.25) \) to get the probability \( P(0 < z < 2.25) = \Phi(2.25) - 0.5 \). This will give you the desired probability.