1. **Problem Statement:** Given Susan Dean's spring pre-calculus exam scores: $$33,42,49,49,53,55,55,63,67,68,68,69,69,78,80,83,88,88,88,60$$ We will find percentiles, dispersion measures, frequency distribution, means, median, mode, and construct a box-whisker plot. --- 2. **Sorting Data:** Sorted data: $$33,42,49,49,53,55,55,60,63,67,68,68,69,69,78,80,83,88,88,88$$ --- 3. **a. Percentiles and Quartiles:** - Number of data points, $$n = 20$$ - 75th percentile (P75): position $$= 0.75(n+1) = 0.75 \times 21 = 15.75$$ Interpolation between 15th and 16th values: 15th = 78, 16th = 80 $$P75 = 78 + 0.75(80-78) = 78 + 1.5 = 79.5$$ - 3rd Decile (D3) = 30th percentile = position $$0.3 \times 21 = 6.3$$ Between 6th and 7th values: 6th = 55, 7th = 55 $$D3 = 55 + 0.3(55-55) = 55$$ - Second quartile (Q2) = median = position $$0.5 \times 21 = 10.5$$ Between 10th and 11th: 10th=67, 11th=68 $$Q2 = 67 + 0.5(68-67) = 67.5$$ Interpretation: P75 = 79.5 means 75% scored below about 80. D3 = 55 means 30% scored below 55. Q2 = 67.5 is the median score. --- 4. **b. Absolute Measures of Dispersion:** - Range: $$\max - \min = 88 - 33 = 55$$ - Interquartile Range (IQR): $$Q3 - Q1$$ Find Q1 and Q3 positions: Q1(position = 0.25*21 = 5.25): 5th=53,6th=55 $$Q1 = 53 + 0.25(55-53) = 53.5$$ Q3(position = 0.75*21 = 15.75) as above = 79.5 $$IQR = 79.5 - 53.5=26$$ - Mean Absolute Deviation (MAD): Calculate mean first: $$\bar{x} = \frac{33+42+49+49+53+55+55+60+63+67+68+68+69+69+78+80+83+88+88+88}{20} = \frac{1331}{20} = 66.55$$ Calculate MAD: $$MAD = \frac{1}{20} \sum |x_i - \bar{x}| = \frac{590.9}{20} = 29.545$$ --- 5. **c. Frequency Distribution:** Choose intervals of width 10: Intervals: 30–39,40–49,50–59,60–69,70–79,80–89 Frequencies: 30–39:1 40–49:3 50–59:3 60–69:7 70–79:2 80–89:4 Relative frequencies: $$f_r = \frac{frequency}{20}$$ 30–39:0.05 40–49:0.15 50–59:0.15 60–69:0.35 70–79:0.10 80–89:0.20 Cumulative relative frequencies: 30–39:0.05 40–49:0.20 50–59:0.35 60–69:0.70 70–79:0.80 80–89:1.00 (Graph omitted since not requested explicitly but description provided) --- 6. **d. Means and Modes for Grouped Data:** Class midpoints: 35, 45, 55, 65, 75, 85 Use frequencies from (c). - Arithmetic mean: $$\bar{x} = \frac{\sum f_i m_i}{n} = \frac{1\times35 + 3\times45 + 3\times55 + 7\times65 + 2\times75 + 4\times85}{20} = \frac{35 + 135 + 165 + 455 + 150 + 340}{20} = \frac{1280}{20} = 64$$ - Geometric mean: $$GM = \left(\prod m_i^{f_i}\right)^{1/n} = \exp\left(\frac{1}{20} \sum f_i \ln m_i\right)$$ Calculate: $$\sum f_i \ln m_i = 1\ln35 + 3\ln45 + 3\ln55 + 7\ln65 + 2\ln75 + 4\ln85 \\ = 3.555 + 10.988 + 12.012 + 28.171 + 8.572 + 17.544 = 80.842$$ $$GM = e^{80.842/20} = e^{4.0421} = 56.96$$ - Harmonic mean: $$HM = \frac{n}{\sum \frac{f_i}{m_i}} = \frac{20}{\frac{1}{35} + \frac{3}{45} + \frac{3}{55} + \frac{7}{65} + \frac{2}{75} + \frac{4}{85}}$$ Calculate denominator: $$= 0.02857 + 0.06667 + 0.05454 + 0.10769 + 0.02667 + 0.04706 = 0.3312$$ $$HM = \frac{20}{0.3312} = 60.39$$ - Median and Mode: Median class: cumulative frequency crosses 10 at 60–69 interval, median = 65 (midpoint) Mode class: highest frequency = 7 in 60–69 interval mode approx 65 --- 7. **e. Box-Whisker Plot:** - Minimum = 33 - Q1 ≈ 53.5 - Median (Q2) = 67.5 - Q3 = 79.5 - Maximum = 88 Draw box from 53.5 to 79.5 with line at 67.5, whiskers to 33 and 88. --- **Final answers:** - P75 = 79.5, D3 = 55, Q2 = 67.5 - Range = 55, IQR = 26, MAD ≈ 29.545 - Frequency distribution and relative frequencies as above - Means: Arithmetic = 64, Geometric ≈ 56.96, Harmonic ≈ 60.39 - Median ≈ 65, Mode ≈ 65 - Box plot values noted