1. **Problem Statement:** We want to compute probabilities and percentiles using the standard normal distribution, which is a normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$. 2. **Formula and Rules:** The standard normal variable is denoted by $Z$. Probabilities are found using the cumulative distribution function (CDF) $P(Z \leq z)$, which is tabulated in the standard normal table. 3. To find the probability that $Z$ is less than or equal to a value $z$, look up $z$ in the table to get $P(Z \leq z)$. 4. To find the probability that $Z$ is greater than $z$, use $P(Z > z) = 1 - P(Z \leq z)$. 5. To find the percentile corresponding to a probability $p$, find the $z$-value such that $P(Z \leq z) = p$ by looking up $p$ in the table and finding the corresponding $z$. 6. **Example:** Find $P(Z \leq 1.25)$. 7. Look up $1.25$ in the standard normal table, which gives approximately $0.8944$. 8. So, $P(Z \leq 1.25) = 0.8944$, meaning there is an 89.44% chance that $Z$ is less than or equal to 1.25. 9. **Example:** Find $P(Z > 1.25)$. 10. Use $P(Z > 1.25) = 1 - P(Z \leq 1.25) = 1 - 0.8944 = 0.1056$. 11. **Example:** Find the $z$-value for the 90th percentile. 12. Look up $0.9000$ in the table to find the closest probability, which corresponds to $z \approx 1.28$. 13. So, the 90th percentile corresponds to $z = 1.28$. This method allows you to compute probabilities and percentiles using the standard normal table effectively.