1. **Problem Statement:** We have frequency distributions of masses (in grams) for black and brown rat colonies. 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 justify. - d) Assess the effect of adding two black rats (330g and 210g) on the median interval. 2. **Formulas and Important Rules:** - Mean: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is frequency and $x_i$ is midpoint. - Median: The middle value when data is ordered; for grouped data, find cumulative frequencies and locate median class. - IQR: $Q_3 - Q_1$, where $Q_1$ and $Q_3$ are first and third quartiles. - SD: $$s = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i - 1}}$$ - Boxplot uses min, $Q_1$, median, $Q_3$, max. 3. **Calculate Midpoints and Frequencies:** Mass intervals and midpoints $x_i$: - 100-150: 125 - 150-200: 175 - 200-250: 225 - 250-300: 275 - 300-350: 325 - 350-400: 375 Black rats frequencies $f_b$: 2,7,18,24,13,6 Brown rats frequencies $f_r$: 1,8,32,22,7,0 4. **Calculate totals:** - Black total $N_b = 2+7+18+24+13+6=70$ - Brown total $N_r = 1+8+32+22+7+0=70$ 5. **Calculate means:** $$\bar{x}_b = \frac{2\times125 + 7\times175 + 18\times225 + 24\times275 + 13\times325 + 6\times375}{70} = \frac{250 + 1225 + 4050 + 6600 + 4225 + 2250}{70} = \frac{18600}{70} = 265.71$$ $$\bar{x}_r = \frac{1\times125 + 8\times175 + 32\times225 + 22\times275 + 7\times325 + 0}{70} = \frac{125 + 1400 + 7200 + 6050 + 2275}{70} = \frac{17050}{70} = 243.57$$ 6. **Calculate cumulative frequencies for median:** Black cumulative: 2, 9, 27, 51, 64, 70 Brown cumulative: 1, 9, 41, 63, 70, 70 Median position: $\frac{N+1}{2} = \frac{70+1}{2} = 35.5$ - Black median class: cumulative just above 35.5 is 51 (class 250-300) - Brown median class: cumulative just above 35.5 is 41 (class 200-250) 7. **Calculate median for grouped data:** Formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times w$$ Where: - $L$ = lower class boundary of median class - $N$ = total frequency - $F$ = cumulative frequency before median class - $f$ = frequency of median class - $w$ = class width For black rats: - $L=249.5$, $N=70$, $F=27$, $f=24$, $w=50$ $$\text{Median}_b = 249.5 + \left(\frac{35 - 27}{24}\right) \times 50 = 249.5 + \frac{8}{24} \times 50 = 249.5 + 16.67 = 266.17$$ For brown rats: - $L=199.5$, $N=70$, $F=9$, $f=32$, $w=50$ $$\text{Median}_r = 199.5 + \left(\frac{35 - 9}{32}\right) \times 50 = 199.5 + \frac{26}{32} \times 50 = 199.5 + 40.63 = 240.13$$ 8. **Calculate quartiles similarly:** - $Q_1$ position = $0.25 \times 70 = 17.5$ - $Q_3$ position = $0.75 \times 70 = 52.5$ Black rats: - $Q_1$ class: cumulative just above 17.5 is 27 (class 200-250) - $Q_3$ class: cumulative just above 52.5 is 64 (class 300-350) Calculate $Q_1$: - $L=199.5$, $F=9$, $f=18$, $w=50$ $$Q_1 = 199.5 + \left(\frac{17.5 - 9}{18}\right) \times 50 = 199.5 + \frac{8.5}{18} \times 50 = 199.5 + 23.61 = 223.11$$ Calculate $Q_3$: - $L=299.5$, $F=51$, $f=13$, $w=50$ $$Q_3 = 299.5 + \left(\frac{52.5 - 51}{13}\right) \times 50 = 299.5 + \frac{1.5}{13} \times 50 = 299.5 + 5.77 = 305.27$$ Brown rats: - $Q_1$ class: cumulative just above 17.5 is 41 (class 200-250) - $Q_3$ class: cumulative just above 52.5 is 63 (class 250-300) Calculate $Q_1$: - $L=199.5$, $F=9$, $f=32$, $w=50$ $$Q_1 = 199.5 + \left(\frac{17.5 - 9}{32}\right) \times 50 = 199.5 + \frac{8.5}{32} \times 50 = 199.5 + 13.28 = 212.78$$ Calculate $Q_3$: - $L=249.5$, $F=41$, $f=22$, $w=50$ $$Q_3 = 249.5 + \left(\frac{52.5 - 41}{22}\right) \times 50 = 249.5 + \frac{11.5}{22} \times 50 = 249.5 + 26.14 = 275.64$$ 9. **Calculate IQR:** - Black rats: $IQR_b = 305.27 - 223.11 = 82.16$ - Brown rats: $IQR_r = 275.64 - 212.78 = 62.86$ 10. **Calculate standard deviations:** Calculate $\sum f_i x_i^2$ for black rats: $$2\times125^2 + 7\times175^2 + 18\times225^2 + 24\times275^2 + 13\times325^2 + 6\times375^2 = 2\times15625 + 7\times30625 + 18\times50625 + 24\times75625 + 13\times105625 + 6\times140625$$ $$= 31250 + 214375 + 911250 + 1815000 + 1373125 + 843750 = 5196750$$ Variance black rats: $$s_b^2 = \frac{5196750 - \frac{(18600)^2}{70}}{69} = \frac{5196750 - \frac{345960000}{70}}{69} = \frac{5196750 - 4942285.71}{69} = \frac{254464.29}{69} = 3687.43$$ $$s_b = \sqrt{3687.43} = 60.72$$ Calculate $\sum f_i x_i^2$ for brown rats: $$1\times125^2 + 8\times175^2 + 32\times225^2 + 22\times275^2 + 7\times325^2 + 0 = 15625 + 8\times30625 + 32\times50625 + 22\times75625 + 7\times105625$$ $$= 15625 + 245000 + 1620000 + 1663750 + 739375 = 4292750$$ Variance brown rats: $$s_r^2 = \frac{4292750 - \frac{(17050)^2}{70}}{69} = \frac{4292750 - \frac{290702500}{70}}{69} = \frac{4292750 - 4152892.86}{69} = \frac{139857.14}{69} = 2026.53$$ $$s_r = \sqrt{2026.53} = 45.02$$ 11. **Spread comparison:** - Black rats have larger SD (60.72 vs 45.02) and larger IQR (82.16 vs 62.86), so black rats have greater spread. 12. **Overweight rats:** - Black rat overweight if mass $> \bar{x}_b + 0.70 s_b = 265.71 + 0.70 \times 60.72 = 265.71 + 42.50 = 308.21$ - Brown rat overweight if mass $> \bar{x}_r + 1.35 s_r = 243.57 + 1.35 \times 45.02 = 243.57 + 60.78 = 304.35$ Count black rats overweight: Classes with midpoint $> 308.21$ are 325 and 375. Frequencies: 13 + 6 = 19 Count brown rats overweight: Classes with midpoint $> 304.35$ are 325 only. Frequency: 7 Black colony has more overweight rats (19 vs 7). 13. **Effect of adding two black rats (330g and 210g):** New total black rats: 72 New cumulative frequencies: - 100-150: 2 - 150-200: 7 + 1 (210g in 200-250?) Actually 210g is in 200-250 class, so frequency in 200-250 increases from 18 to 19 - 300-350: 13 + 1 (330g in 300-350) increases to 14 New frequencies: 2,7,19,24,14,6 New cumulative: 2,9,28,52,66,72 Median position: $\frac{72+1}{2} = 36.5$ Median class still 250-300 (cumulative 52 > 36.5) Since median class unchanged, median interval does not change. **David's claim is incorrect.**