1. **Problem Statement:** We are given a set of times (in seconds) taken by employees to complete a task: 63, 229, 165, 77, 49, 74, 67, 59, 66, 102, 81, 72, 59, 74, 61, 82, 48, 70, 86. 2. **Stem-and-Leaf Plot:** - The stem represents the tens digit, the leaf the units digit. - Data sorted: 48, 49, 59, 59, 61, 63, 66, 67, 70, 72, 74, 74, 77, 81, 82, 86, 102, 165, 229. - Stem | Leaf 4 | 8 9 5 | 9 9 6 | 1 3 6 7 7 | 0 2 4 4 7 8 | 1 2 6 10 | 2 16 | 5 22 | 9 This shows most times are between 40 and 90 seconds, with a few outliers at 102, 165, and 229. 3. **Arithmetic Mean:** - Formula: $\bar{x} = \frac{\sum x_i}{n}$ - Sum: $48 + 49 + 59 + 59 + 61 + 63 + 66 + 67 + 70 + 72 + 74 + 74 + 77 + 81 + 82 + 86 + 102 + 165 + 229 = 1595$ - Number of data points $n=19$ - Mean: $\frac{1595}{19} = 83.95$ 4. **Mode:** - The value(s) that appear most frequently. - 59 and 74 appear twice each, others less. - Modes: 59 and 74 (bimodal) 5. **Median:** - Middle value when data is sorted. - Position: $\frac{n+1}{2} = 10^{th}$ value. - Sorted data 10th value: 72 - Median = 72 6. **Quartiles:** - Q1 (25th percentile): median of first 9 values: 59 - Q3 (75th percentile): median of last 9 values: 81 7. **80th Percentile:** - Position: $0.8 \times (n+1) = 0.8 \times 20 = 16^{th}$ value - 16th value in sorted data: 86 - Meaning: 80% of employees took 86 seconds or less. 8. **Variance and Standard Deviation:** - Variance formula: $s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$ - Calculate squared deviations and sum: $$\sum (x_i - 83.95)^2 = 20488.95$$ - Variance: $\frac{20488.95}{18} = 1138.27$ - Standard deviation: $s = \sqrt{1138.27} = 33.74$ 9. **Coefficient of Variation (CV):** - $CV = \frac{s}{\bar{x}} = \frac{33.74}{83.95} = 0.402$ or 40.2% - CV measures relative variability; important to compare variability between datasets with different units or means. 10. **Box-Plot Construction:** - Minimum: 48 - Q1: 59 - Median: 72 - Q3: 81 - Maximum: 229 - Outliers: Values beyond $Q3 + 1.5 \times IQR$ or below $Q1 - 1.5 \times IQR$ - IQR = $81 - 59 = 22$ - Upper fence: $81 + 1.5 \times 22 = 114$ - Lower fence: $59 - 1.5 \times 22 = 26$ - Outliers: 165 and 229 (above 114) 11. **Skewness and Outliers:** - Data is right-skewed (positive skew) due to high outliers. - Outliers identified: 165 and 229. **Final answers:** - Mean = 83.95 - Mode = 59, 74 - Median = 72 - Q1 = 59, Q3 = 81 - 80th percentile = 86 - Variance = 1138.27 - Standard deviation = 33.74 - Coefficient of variation = 0.402 - Data is right-skewed with outliers 165 and 229