1. **Problem Statement:** We have a sample 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:** - Stem represents the tens digit, leaf the units digit. - Sorted data: 48, 49, 59, 59, 61, 63, 66, 67, 70, 72, 74, 74, 77, 81, 82, 86, 102, 165, 229. - Plot: 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 - From this, we see most data is between 40 and 90 seconds, with some high outliers (102, 165, 229). 3. **Measures of Central Tendency and Quartiles:** - Mean: $\frac{63+229+165+77+49+74+67+59+66+102+81+72+59+74+61+82+48+70+86}{19} = \frac{1594}{19} \approx 83.89$ - Mode: 59 and 74 (both appear twice) - Median: Middle value of sorted data (19 values), 10th value = 72 - Quartiles: - Q1 (25th percentile): median of first 9 values = 61 - Q3 (75th percentile): median of last 9 values = 82 4. **80th Percentile:** - Position = $0.8 \times (19+1) = 16$th value in sorted data = 86 - Meaning: 80% of employees took 86 seconds or less to complete the task. 5. **Variance and Standard Deviation:** - Variance $s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$ - Calculate squared deviations, sum them, divide by 18: Sum of squared deviations $\approx 20494.32$ Variance $= \frac{20494.32}{18} \approx 1138.57$ - Standard deviation $s = \sqrt{1138.57} \approx 33.74$ 6. **Coefficient of Variation (CV):** - $CV = \frac{s}{\bar{x}} = \frac{33.74}{83.89} \approx 0.402$ - CV measures relative variability; important to compare variability between datasets with different units or means. 7. **Box-Plot Construction:** - Minimum: 48 - Q1: 61 - Median: 72 - Q3: 82 - Maximum: 229 - Outliers: Values > $Q3 + 1.5 \times IQR = 82 + 1.5 \times (82-61) = 82 + 31.5 = 113.5$ - Outliers are 165 and 229. - Box-plot shows most data concentrated between 48 and 82 with some high outliers. 8. **Skewness and Outliers:** - Data is right-skewed due to high outliers (165, 229). - Outliers identified above. Final answers: - Mean $\approx 83.89$ - Mode = 59, 74 - Median = 72 - Q1 = 61, Q3 = 82 - 80th percentile = 86 - Variance $\approx 1138.57$ - Standard deviation $\approx 33.74$ - Coefficient of variation $\approx 0.402$ - Data is right-skewed with outliers at 165 and 229.