1. **Problem Statement:** We have wait times at a coffee shop that follow a normal distribution with mean $\mu=8$ minutes and standard deviation $\sigma=2$ minutes. We want to explain the key properties of the normal distribution and calculate the Z-score and probability that the wait time exceeds 10 minutes. 2. **Key Properties of the Normal Distribution:** - The distribution is symmetric around the mean $\mu$. - About 68% of values lie within one standard deviation ($\mu \pm \sigma$). - About 95% lie within two standard deviations ($\mu \pm 2\sigma$). - About 99.7% lie within three standard deviations ($\mu \pm 3\sigma$). **Practical Benefit:** This helps predict typical wait times and prepare for rare long waits. For example, most waits will be between 6 and 10 minutes (8 $\pm$ 2), so you can expect a short wait usually but be ready for occasional longer waits. 3. **Formula for Z-score:** $$Z = \frac{X - \mu}{\sigma}$$ where $X$ is the observed value. 4. **Calculate Z-score for $X=10$ minutes:** $$Z = \frac{10 - 8}{2} = \frac{2}{2} = 1$$ 5. **Find Probability $P(X > 10)$:** - Using the standard normal table, $P(Z < 1) = 0.8413$ (area to the left). - So, $P(Z > 1) = 1 - 0.8413 = 0.1587$. **Interpretation:** There is about a 15.87% chance the wait time exceeds 10 minutes on any given day. This analysis helps you understand and anticipate wait times using the normal distribution and Z-scores.