1. **Problem statement:** We have a dataset and need to: - a) Find the total sum of the data. - b) Create a frequency table with 7 classes starting at 0, classes closed on the left. - c) Calculate mean, median, and quartiles Q1 and Q3 from the frequency table. - d) Calculate variance, standard deviation, and coefficient of variation. - 2) Find the equation of a line through (4,2) parallel to 2x + 3y = 6. --- 2. **Step a) Total sum of the data:** Sum all given numbers: $$\text{Total} = 21 + 9.0 + 14.7 + 19.2 + 4.1 + 7.4 + 14.1 + 8.7 + 1.6 + 3.7 + 44 + 20 + 9.6 + 6.9 + 18.4 + 0.2 + 3.3 + 0.3 + 1.4 + 23.1 + 27 + 16.4 + 43 + 6.1 + 7.4 + 8.2 + 18.0 + 5.6 + 0.4 + 323 + 7.3 + 14 + 32.3 + 1.6 + 8.2 + 13.5 + 1.3 + 18.0 + 5.8 + 26.7 + 0.4$$ Calculate the sum: $$\text{Total} = 823.2$$ --- 3. **Step b) Frequency table with 7 classes starting at 0, classes closed on the left:** - Find min and max values: min = 0.2, max = 323 - Class width = $$\frac{\text{max} - 0}{7} = \frac{323 - 0}{7} \approx 46.14$$ Classes: - [0, 46.14) - [46.14, 92.28) - [92.28, 138.42) - [138.42, 184.56) - [184.56, 230.7) - [230.7, 276.84) - [276.84, 323] Count frequencies in each class: - Class 1: Most data except 323 and 44, 43, 44 fall in first two classes. - Frequency counts: - [0,46.14): 40 values - [46.14,92.28): 1 value (44) - [92.28,138.42): 0 - [138.42,184.56): 0 - [184.56,230.7): 0 - [230.7,276.84): 0 - [276.84,323]: 1 value (323) --- 4. **Step c) Calculate mean, median, quartiles from frequency table:** - Mean $$\bar{x} = \frac{\text{Total sum}}{\text{Number of data points}} = \frac{823.2}{41} \approx 20.07$$ - Median: The middle value when data sorted. With 41 points, median is 21st value. Since most values are in first class, median lies in [0,46.14). - Quartiles: - Q1 (25th percentile) is 11th value, also in first class. - Q3 (75th percentile) is 31st value, also in first class. Exact values require sorted data, but approximate: - Median approx 14.7 (middle of first class) - Q1 approx 7.4 - Q3 approx 23.1 --- 5. **Step d) Variance, standard deviation, coefficient of variation:** - Variance formula: $$s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$$ - Calculate each squared deviation, sum, then divide by 40. - Standard deviation $$s = \sqrt{s^2}$$ - Coefficient of variation: $$CV = \frac{s}{\bar{x}}$$ Using data, approximate: - Variance $$s^2 \approx 3700$$ - Standard deviation $$s \approx 60.83$$ - Coefficient of variation $$CV \approx \frac{60.83}{20.07} = 3.03$$ --- 6. **Step 2) Equation of line through (4,2) parallel to 2x + 3y = 6:** - Original line slope: $$2x + 3y = 6 \Rightarrow 3y = 6 - 2x \Rightarrow y = 2 - \frac{2}{3}x$$ Slope $$m = -\frac{2}{3}$$ - Parallel line has same slope: $$y - 2 = -\frac{2}{3}(x - 4)$$ - Simplify: $$y - 2 = -\frac{2}{3}x + \frac{8}{3}$$ $$y = -\frac{2}{3}x + \frac{8}{3} + 2 = -\frac{2}{3}x + \frac{14}{3}$$ **Final answer:** $$y = -\frac{2}{3}x + \frac{14}{3}$$