1. **Stating the problem:** We are asked to compute the mean, median, P30, P60, P90, Q1, Q3, range, and interquartile range for the sales data of the top 15 market research firms in the U.S. in 2018. 2. **Data given:** Sales in millions: 3730, 1430, 970, 733, 569, 551, 332, 300, 289, 171, 167, 133, 130, 124, 108, 108. 3. **Sort the data in ascending order:** $$108, 108, 124, 130, 133, 167, 171, 289, 300, 332, 551, 569, 733, 970, 1430, 3730$$ 4. **Count the number of data points:** There are $n=16$ values. 5. **Mean formula:** $$\text{Mean} = \frac{\sum x_i}{n}$$ Calculate sum: $$108 + 108 + 124 + 130 + 133 + 167 + 171 + 289 + 300 + 332 + 551 + 569 + 733 + 970 + 1430 + 3730 = 8845$$ So, $$\text{Mean} = \frac{8845}{16} = 552.8125$$ 6. **Median:** For even $n$, median is average of middle two values at positions $\frac{n}{2}$ and $\frac{n}{2}+1$, i.e., 8th and 9th values. Values at 8th and 9th positions: 289 and 300. $$\text{Median} = \frac{289 + 300}{2} = 294.5$$ 7. **Percentiles:** Use the formula for the $k$th percentile position: $$P_k = \text{value at } L = \frac{k}{100}(n+1)$$ - $P_{30}$ position: $$L = 0.30 \times 17 = 5.1$$ Value at 5th position is 133, 6th is 167. Interpolate: $$P_{30} = 133 + 0.1 \times (167 - 133) = 133 + 3.4 = 136.4$$ - $P_{60}$ position: $$L = 0.60 \times 17 = 10.2$$ Value at 10th is 332, 11th is 551. Interpolate: $$P_{60} = 332 + 0.2 \times (551 - 332) = 332 + 43.8 = 375.8$$ - $P_{90}$ position: $$L = 0.90 \times 17 = 15.3$$ Value at 15th is 1430, 16th is 3730. Interpolate: $$P_{90} = 1430 + 0.3 \times (3730 - 1430) = 1430 + 870 = 2300$$ 8. **Quartiles:** - $Q_1 = P_{25}$ position: $$L = 0.25 \times 17 = 4.25$$ Value at 4th is 130, 5th is 133. Interpolate: $$Q_1 = 130 + 0.25 \times (133 - 130) = 130 + 0.75 = 130.75$$ - $Q_3 = P_{75}$ position: $$L = 0.75 \times 17 = 12.75$$ Value at 12th is 569, 13th is 733. Interpolate: $$Q_3 = 569 + 0.75 \times (733 - 569) = 569 + 123 = 692$$ 9. **Range:** $$\text{Range} = \text{max} - \text{min} = 3730 - 108 = 3622$$ 10. **Interquartile Range (IQR):** $$\text{IQR} = Q_3 - Q_1 = 692 - 130.75 = 561.25$$ **Final answers:** - Mean = 552.81 - Median = 294.5 - P30 = 136.4 - P60 = 375.8 - P90 = 2300 - Q1 = 130.75 - Q3 = 692 - Range = 3622 - Interquartile Range = 561.25