1. **State the problem:** We have mass frequency data for black and brown rats in intervals. We need to: - a) Draw boxplots for both data sets. - b) Calculate mean, median, interquartile range (IQR), and standard deviation (SD) for both. - c) Determine which rat type has greater spread and find which colony has more overweight rats based on given SD thresholds. 2. **Calculate midpoints for each mass interval:** - Midpoints $m_i$ are the average of interval bounds: - 100–150: $\frac{100+150}{2} = 125$ - 150–200: $175$ - 200–250: $225$ - 250–300: $275$ - 300–350: $325$ - 350–400: $375$ 3. **Calculate total frequencies:** - Black rats total $N_b = 2+7+18+24+13+6 = 70$ - Brown rats total $N_r = 1+8+32+22+7+0 = 70$ 4. **Calculate mean for each:** - Mean formula: $$\bar{x} = \frac{\sum f_i m_i}{N}$$ - Black rats mean: $$\bar{x}_b = \frac{2\times125 + 7\times175 + 18\times225 + 24\times275 + 13\times325 + 6\times375}{70}$$ $$= \frac{250 + 1225 + 4050 + 6600 + 4225 + 2250}{70} = \frac{18600}{70} = 265.71$$ - Brown rats mean: $$\bar{x}_r = \frac{1\times125 + 8\times175 + 32\times225 + 22\times275 + 7\times325 + 0\times375}{70}$$ $$= \frac{125 + 1400 + 7200 + 6050 + 2275 + 0}{70} = \frac{17050}{70} = 243.57$$ 5. **Calculate median:** - Median position: $\frac{N+1}{2} = \frac{70+1}{2} = 35.5$th value - Black rats cumulative frequencies: - 2, 9, 27, 51, 64, 70 Median lies in 250–300 interval (since 35.5 is between 27 and 51) Median formula in grouped data: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times w$$ Where: - $L=250$ (lower bound of median class) - $N=70$ - $F=27$ (cumulative freq before median class) - $f=24$ (freq of median class) - $w=50$ (class width) $$= 250 + \left(\frac{35 - 27}{24}\right) \times 50 = 250 + \frac{8}{24} \times 50 = 250 + 16.67 = 266.67$$ - Brown rats cumulative frequencies: - 1, 9, 41, 63, 70, 70 Median lies in 200–250 interval (since 35.5 is between 9 and 41) $$L=200, F=9, f=32, w=50$$ $$= 200 + \left(\frac{35 - 9}{32}\right) \times 50 = 200 + \frac{26}{32} \times 50 = 200 + 40.63 = 240.63$$ 6. **Calculate quartiles (Q1 and Q3) for IQR:** - Positions: - $Q1 = \frac{N+1}{4} = 17.75$th value - $Q3 = 3\times\frac{N+1}{4} = 53.25$th value - Black rats Q1: - Cumulative freq before 150–200 is 2 - 150–200 freq = 7 - $L=150, F=2, f=7, w=50$ $$Q1 = 150 + \left(\frac{17.75 - 2}{7}\right) \times 50 = 150 + \frac{15.75}{7} \times 50 = 150 + 112.5 = 262.5$$ - Black rats Q3: - Cumulative freq before 300–350 is 51 - 300–350 freq = 13 - $L=300, F=51, f=13, w=50$ $$Q3 = 300 + \left(\frac{53.25 - 51}{13}\right) \times 50 = 300 + \frac{2.25}{13} \times 50 = 300 + 8.65 = 308.65$$ - Brown rats Q1: - Cumulative freq before 150–200 is 1 - 150–200 freq = 8 - $L=150, F=1, f=8, w=50$ $$Q1 = 150 + \left(\frac{17.75 - 1}{8}\right) \times 50 = 150 + \frac{16.75}{8} \times 50 = 150 + 104.69 = 254.69$$ - Brown rats Q3: - Cumulative freq before 300–350 is 63 - 300–350 freq = 7 - $L=300, F=63, f=7, w=50$ $$Q3 = 300 + \left(\frac{53.25 - 63}{7}\right) \times 50$$ Since 53.25 < 63, Q3 lies in 250–300 interval: - Cumulative freq before 250–300 is 41 - 250–300 freq = 22 - $L=250, F=41, f=22, w=50$ $$Q3 = 250 + \left(\frac{53.25 - 41}{22}\right) \times 50 = 250 + \frac{12.25}{22} \times 50 = 250 + 27.84 = 277.84$$ - Calculate IQR: - Black rats: $IQR_b = Q3 - Q1 = 308.65 - 262.5 = 46.15$ - Brown rats: $IQR_r = 277.84 - 254.69 = 23.15$ 7. **Calculate standard deviation (SD):** - Use formula: $$s = \sqrt{\frac{\sum f_i (m_i - \bar{x})^2}{N}}$$ - Calculate for black rats: $$\sum f_i (m_i - \bar{x}_b)^2 = 2(125-265.71)^2 + 7(175-265.71)^2 + 18(225-265.71)^2 + 24(275-265.71)^2 + 13(325-265.71)^2 + 6(375-265.71)^2$$ $$= 2(18398.7) + 7(8284.5) + 18(1673.5) + 24(86.3) + 13(3520.7) + 6(12192.7)$$ $$= 36797.4 + 57991.5 + 30123 + 2071.2 + 45769.1 + 73156.2 = 246908.4$$ $$s_b = \sqrt{\frac{246908.4}{70}} = \sqrt{3527.26} = 59.39$$ - Calculate for brown rats: $$\sum f_i (m_i - \bar{x}_r)^2 = 1(125-243.57)^2 + 8(175-243.57)^2 + 32(225-243.57)^2 + 22(275-243.57)^2 + 7(325-243.57)^2 + 0(375-243.57)^2$$ $$= 1(14088.4) + 8(4696.5) + 32(345.1) + 22(992.3) + 7(6607.3) + 0$$ $$= 14088.4 + 37572 + 11043.2 + 21830.6 + 46251.1 = 130785.3$$ $$s_r = \sqrt{\frac{130785.3}{70}} = \sqrt{1868.36} = 43.23$$ 8. **Determine which rat type has greater spread:** - Black rats SD = 59.39, Brown rats SD = 43.23 - Black rats have greater spread of masses. 9. **Determine overweight rats:** - Black rat overweight if mass $> \bar{x}_b + 0.95 s_b = 265.71 + 0.95 \times 59.39 = 265.71 + 56.42 = 322.13$ - Brown rat overweight if mass $> \bar{x}_r + 1.38 s_r = 243.57 + 1.38 \times 43.23 = 243.57 + 59.67 = 303.24$ 10. **Count overweight rats:** - Black rats mass intervals above 322.13 are 325–375 and 375–400: - 300–350 freq = 13 (partially overweight) - 350–400 freq = 6 Approximate overweight in 300–350: - Portion overweight in 300–350 interval = $\frac{325-350}{50} = \frac{25}{50} = 0.5$ (half) - Overweight in 300–350 = $13 \times 0.5 = 6.5$ Total overweight black rats = $6.5 + 6 = 12.5 \approx 13$ - Brown rats mass intervals above 303.24 are 325–350 and 350–400: - 300–350 freq = 7 (partially overweight) - 350–400 freq = 0 Portion overweight in 300–350 = $\frac{325-350}{50} = 0.5$ Overweight in 300–350 = $7 \times 0.5 = 3.5$ Total overweight brown rats = $3.5 + 0 = 3.5 \approx 4$ 11. **Conclusion:** - Black rats have greater spread. - Black rat colony contains more overweight rats (about 13) than brown rat colony (about 4). **Final answers:** - Black rats mean = 265.71, median = 266.67, IQR = 46.15, SD = 59.39 - Brown rats mean = 243.57, median = 240.63, IQR = 23.15, SD = 43.23 - Greater spread: Black rats - More overweight rats: Black rats colony