1. **Stating the problem:** We are working with the standard normal distribution, which has mean $\mu=0$ and standard deviation $\sigma=1$. We use the Z-score formula to convert any normal distribution value $X$ to the standard normal distribution: $$Z = \frac{X - \mu}{\sigma}$$ This formula tells us how many standard deviations a value $X$ is from the mean. 2. **Properties and rules:** - The normal curve is bell-shaped and symmetric about the mean. - The total area under the curve is 1, representing total probability. - Empirical rule: - About 68.26% of data lies within $\pm 1\sigma$ - About 95.44% within $\pm 2\sigma$ - About 99.74% within $\pm 3\sigma$ 3. **Example calculations:** - For a value $X$, calculate $Z$ using $Z=\frac{X-\mu}{\sigma}$. - Use standard normal tables or empirical rule to find probabilities or proportions. 4. **Example 1:** Given $\mu=60$, $\sigma=5$, find $Z$ for Billy's score $X=65$: $$Z = \frac{65-60}{5} = 1$$ Billy's score is 1 standard deviation above the mean. 5. **Example 2:** Mango orchard yield $\mu=70$, $\sigma=16$. Find probability orchard yield is between 38 and 70: Calculate $Z$ for 38: $$Z = \frac{38-70}{16} = -2$$ For 70: $$Z = \frac{70-70}{16} = 0$$ From empirical rule or Z-tables, area between $Z=-2$ and $Z=0$ is approximately 47.72%. 6. **Example 3:** Problem solving test $\mu=60$, $\sigma=9$, $n=50$. Find percentage between 60 and 69: Calculate $Z$ for 69: $$Z = \frac{69-60}{9} = 1$$ Area between mean and $Z=1$ is about 34.13%. 7. **Problem Set 1 (selected):** a. Proportion between mean and $+2.3\sigma$: From Z-tables, area between 0 and 2.3 is approximately 48.93%. b. Proportion below $-1.2\sigma$: From Z-tables, area below $-1.2$ is about 11.51%. c. Proportion between $0.25\sigma$ and $+0.6\sigma$: Area between Z=0.25 and Z=0.6 is about 13.11%. d. Proportion between $+2.0\sigma$ and $0.2\sigma$: Area between Z=0.2 and Z=2.0 is about 41.59%. e. Proportion below $0.3\sigma$: Area below Z=0.3 is about 61.79%. 8. **Summary:** Use the Z-score formula to convert values to standard normal distribution, then use Z-tables or empirical rule to find probabilities or proportions under the curve. This helps in understanding how data is distributed relative to the mean in any normal distribution.