1. **Problem 1:** Given $x=78$, $\mu=70$, $\sigma=4$, find the z-score. The formula for the z-score is: $$z=\frac{x-\mu}{\sigma}$$ Substitute the values: $$z=\frac{78-70}{4}$$ Calculate the numerator: $$z=\frac{8}{4}$$ Simplify the fraction: $$z=\frac{\cancel{8}}{\cancel{4}}=2$$ So, the z-score is $2$. 2. **Problem 2:** Given $x=52$, $\mu=60$, $\sigma=8$, find the z-score. Use the formula: $$z=\frac{x-\mu}{\sigma}$$ Substitute values: $$z=\frac{52-60}{8}$$ Calculate numerator: $$z=\frac{-8}{8}$$ Simplify: $$z=\frac{\cancel{-8}}{\cancel{8}}=-1$$ The z-score is $-1$. 3. **Problem 3:** Given $x=95$, $\mu=85$, $\sigma=5$, find the z-score. Formula: $$z=\frac{x-\mu}{\sigma}$$ Substitute: $$z=\frac{95-85}{5}$$ Calculate numerator: $$z=\frac{10}{5}$$ Simplify: $$z=\frac{\cancel{10}}{\cancel{5}}=2$$ The z-score is $2$. 4. **Problem 4:** Given $x=40$, $\mu=50$, $\sigma=10$, find the z-score. Formula: $$z=\frac{x-\mu}{\sigma}$$ Substitute: $$z=\frac{40-50}{10}$$ Calculate numerator: $$z=\frac{-10}{10}$$ Simplify: $$z=\frac{\cancel{-10}}{\cancel{10}}=-1$$ The z-score is $-1$. 5. **Problem 5:** Given $x=110$, $\mu=100$, $\sigma=15$, find the z-score. Formula: $$z=\frac{x-\mu}{\sigma}$$ Substitute: $$z=\frac{110-100}{15}$$ Calculate numerator: $$z=\frac{10}{15}$$ Simplify fraction: $$z=\frac{\cancel{10}}{\cancel{15}}=\frac{2}{3}$$ The z-score is $\frac{2}{3}$ or approximately $0.67$.