1. Stating the problem: We have the data set: 108, 200, 310, 150, 180, 119, 160, 180, 201, 190, 280, 202, and we want to convert the given values (z-score, population, sample) into percentiles. 2. Understanding terms: - Z-score measures how many standard deviations an element is from the mean. - Percentile rank of a z-score can be found using the standard normal distribution. 3. Given the data values are raw data, we first calculate the mean $$\mu = \frac{\sum x_i}{n}$$ and standard deviation $$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$$ for population or $$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$$ for sample. 4. Calculate mean $$\mu = \frac{108 + 200 + 310 + 150 + 180 + 119 + 160 + 180 + 201 + 190 + 280 + 202}{12} = \frac{2290}{12} = 190.83$$ 5. Calculate squared deviations: (108-190.83)^2 = 6886.71 (200-190.83)^2 = 84.03 (310-190.83)^2 = 14144.7 (150-190.83)^2 = 1686.97 (180-190.83)^2 = 117.36 (119-190.83)^2 = 5211.7 (160-190.83)^2 = 952.36 (180-190.83)^2 = 117.36 (201-190.83)^2 = 104.69 (190-190.83)^2 = 0.69 (280-190.83)^2 = 7923.36 (202-190.83)^2 = 125.43 6. Sum of squared deviations = 6886.71 + 84.03 + 14144.7 + 1686.97 + 117.36 + 5211.7 + 952.36 + 117.36 + 104.69 + 0.69 + 7923.36 + 125.43 = 36054.36 7. Population standard deviation $$\sigma = \sqrt{\frac{36054.36}{12}} = \sqrt{3004.53} = 54.82$$ 8. Now compute z-score for each data point using $$z = \frac{x - \mu}{\sigma}$$, e.g., for 108 $$z = \frac{108 - 190.83}{54.82} = -1.52$$ 9. Find percentile for each z-score using the standard normal table or cumulative distribution function. For z = -1.52, percentile ≈ 6.43% 10. Repeat for all values to convert them into percentiles. Final answers: example percentiles for first and last values are approximately 6.43% (for 108) and 87.5% (for 202).