1. **Problem Statement:** Find the proportion of the area under the standard normal curve for the given intervals in terms of standard deviations ($\sigma$) from the mean. 2. **Formula and Rules:** The standard normal distribution is symmetric about the mean (0). The cumulative distribution function (CDF), $\Phi(z)$, gives the area to the left of $z$. Use standard normal tables or a calculator for $\Phi(z)$ values. 3. **Calculations:** **a. Between the mean (0) and +2.3$\sigma$:** $$\text{Area} = \Phi(2.3) - \Phi(0)$$ From tables, $\Phi(2.3) \approx 0.9893$ and $\Phi(0) = 0.5$ $$= 0.9893 - 0.5 = 0.4893$$ **b. Below -1.2$\sigma$:** $$\text{Area} = \Phi(-1.2)$$ From tables, $\Phi(-1.2) \approx 0.1151$ **c. Between 0.25$\sigma$ and +0.6$\sigma$:** $$\text{Area} = \Phi(0.6) - \Phi(0.25)$$ From tables, $\Phi(0.6) \approx 0.7257$, $\Phi(0.25) \approx 0.5987$ $$= 0.7257 - 0.5987 = 0.1270$$ **d. Between +2.0$\sigma$ and 0.2$\sigma$:** Note order: lower limit is 0.2, upper is 2.0 $$\text{Area} = \Phi(2.0) - \Phi(0.2)$$ From tables, $\Phi(2.0) \approx 0.9772$, $\Phi(0.2) \approx 0.5793$ $$= 0.9772 - 0.5793 = 0.3979$$ **e. Below 0.3$\sigma$:** $$\text{Area} = \Phi(0.3)$$ From tables, $\Phi(0.3) \approx 0.6179$ 4. **Summary:** - a: 0.4893 - b: 0.1151 - c: 0.1270 - d: 0.3979 - e: 0.6179 These values represent the proportions of the total area under the normal curve for the specified intervals.