1. **Problem:** Find the area under the standard normal distribution curve between $z=0$ and $z=1.5$. 2. **Formula and rules:** The area between two z-scores on the standard normal distribution corresponds to the probability that a value falls between those z-scores. We use the cumulative distribution function (CDF) of the standard normal distribution, denoted as $\Phi(z)$. The area between $z=a$ and $z=b$ is given by: $$\text{Area} = \Phi(b) - \Phi(a)$$ 3. **Step-by-step solution:** - Find $\Phi(1.5)$, the cumulative probability up to $z=1.5$. - Find $\Phi(0)$, the cumulative probability up to $z=0$. From standard normal tables or a calculator: $$\Phi(1.5) \approx 0.9332$$ $$\Phi(0) = 0.5$$ 4. **Calculate the area:** $$\text{Area} = 0.9332 - 0.5 = 0.4332$$ 5. **Interpretation:** The area between $z=0$ and $z=1.5$ under the standard normal curve is approximately $0.4332$, meaning there is a 43.32% probability that a value lies between these z-scores.