1. **State the problem:** We need to find the z-score such that the cumulative probability $P(Z < a) = 0.9911$ for a standard normal distribution. 2. **Recall the definition:** The z-score corresponds to the value on the standard normal curve where the area to the left (cumulative probability) is 0.9911. 3. **Use the standard normal distribution table or inverse function:** We look up the value in the z-table or use a calculator to find $z$ such that $\Phi(z) = 0.9911$. 4. **Find the z-score:** From the z-table or using an inverse normal function, $z \approx 2.33$ because $P(Z < 2.33) \approx 0.9911$. 5. **Interpretation:** This means that about 99.11% of the data lies below a z-score of 2.33 in a standard normal distribution. **Final answer:** $$z = 2.33$$