1. **State the problem:** Calculate the z-score for the measurements 34.6, 35.6, and 37.2 using the formula $$z = \frac{x - \bar{x}}{s}$$ where $\bar{x}$ is the mean and $s$ is the standard deviation. 2. **Calculate the mean $\bar{x}$:** $$\bar{x} = \frac{34.6 + 35.6 + 37.2}{3} = \frac{107.4}{3} = 35.8$$ 3. **Calculate the standard deviation $s$:** First, find the squared differences from the mean: $$ (34.6 - 35.8)^2 = (-1.2)^2 = 1.44 $$ $$ (35.6 - 35.8)^2 = (-0.2)^2 = 0.04 $$ $$ (37.2 - 35.8)^2 = (1.4)^2 = 1.96 $$ Sum these squared differences: $$ 1.44 + 0.04 + 1.96 = 3.44 $$ Divide by $n-1 = 2$ (sample standard deviation): $$ \frac{3.44}{2} = 1.72 $$ Take the square root: $$ s = \sqrt{1.72} \approx 1.311 $$ 4. **Calculate each z-score:** For $x=34.6$: $$ z = \frac{34.6 - 35.8}{1.311} = \frac{-1.2}{1.311} \approx -0.915 $$ For $x=35.6$: $$ z = \frac{35.6 - 35.8}{1.311} = \frac{-0.2}{1.311} \approx -0.153 $$ For $x=37.2$: $$ z = \frac{37.2 - 35.8}{1.311} = \frac{1.4}{1.311} \approx 1.068 $$ 5. **Final answer:** The z-scores are approximately: - For 34.6: $-0.915$ - For 35.6: $-0.153$ - For 37.2: $1.068$