1. **State the problem:** We are given a data set with mean $\mu = 75$ and standard deviation $\sigma = 11.5$. We need to find the z-score for each value $x = 85, 50, 95, 75$. 2. **Formula for z-score:** $$z = \frac{x - \mu}{\sigma}$$ This formula tells us how many standard deviations a value $x$ is from the mean $\mu$. 3. **Calculate each z-score:** - For $x = 85$: $$z = \frac{85 - 75}{11.5} = \frac{10}{11.5} \approx 0.87$$ - For $x = 50$: $$z = \frac{50 - 75}{11.5} = \frac{-25}{11.5} \approx -2.17$$ - For $x = 95$: $$z = \frac{95 - 75}{11.5} = \frac{20}{11.5} \approx 1.74$$ - For $x = 75$: $$z = \frac{75 - 75}{11.5} = \frac{0}{11.5} = 0$$ 4. **Interpretation:** - A positive z-score means the value is above the mean. - A negative z-score means the value is below the mean. - A z-score of 0 means the value is exactly the mean. **Final answers:** - $z_{85} \approx 0.87$ - $z_{50} \approx -2.17$ - $z_{95} \approx 1.74$ - $z_{75} = 0$