1. **Problem Statement:** We have grouped data of weekly registrations with frequencies and cumulative frequencies. We need to construct an ogive, estimate median, quartiles, interquartile range, 60th percentile, and probabilities from the ogive. 2. **Ogive Construction:** The ogive plots cumulative frequency against the upper boundary of each class interval. 3. **Data Points for Ogive:** - (10, 4) - (20, 14) - (30, 32) - (40, 58) - (50, 80) - (60, 100) 4. **Median Estimation:** Median corresponds to cumulative frequency $\frac{100}{2} = 50$. - Locate 50 on y-axis, find corresponding x on ogive. - Between 40 (58) and 30 (32), median lies. - Use linear interpolation: $$x = 30 + \frac{50 - 32}{58 - 32} \times (40 - 30) = 30 + \frac{18}{26} \times 10 = 30 + 6.92 = 36.92$$ 5. **Quartiles:** - First quartile $Q_1$ at cumulative frequency $\frac{100}{4} = 25$. - Third quartile $Q_3$ at cumulative frequency $\frac{3 \times 100}{4} = 75$. - For $Q_1$ between 20 (14) and 30 (32): $$Q_1 = 20 + \frac{25 - 14}{32 - 14} \times (30 - 20) = 20 + \frac{11}{18} \times 10 = 20 + 6.11 = 26.11$$ - For $Q_3$ between 40 (58) and 50 (80): $$Q_3 = 40 + \frac{75 - 58}{80 - 58} \times (50 - 40) = 40 + \frac{17}{22} \times 10 = 40 + 7.73 = 47.73$$ 6. **Interquartile Range (IQR):** $$IQR = Q_3 - Q_1 = 47.73 - 26.11 = 21.62$$ 7. **60th Percentile $P_{60}$:** - At cumulative frequency $60$. - Between 30 (32) and 40 (58): $$P_{60} = 30 + \frac{60 - 32}{58 - 32} \times (40 - 30) = 30 + \frac{28}{26} \times 10 = 30 + 10.77 = 40.77$$ 8. **Probabilities:** Total weeks = 100. - (i) Registrations less than 25: - 25 lies between 20 (14) and 30 (32). - Interpolate cumulative frequency at 25: $$CF_{25} = 14 + \frac{25 - 20}{30 - 20} \times (32 - 14) = 14 + \frac{5}{10} \times 18 = 14 + 9 = 23$$ - Probability = $\frac{23}{100} = 0.23$ - (ii) Registrations between 20 and 45: - CF at 20 = 14, at 45 interpolate between 40 (58) and 50 (80): $$CF_{45} = 58 + \frac{45 - 40}{50 - 40} \times (80 - 58) = 58 + \frac{5}{10} \times 22 = 58 + 11 = 69$$ - Probability = $\frac{69 - 14}{100} = 0.55$ - (iii) Registrations greater than 40: - CF at 40 = 58 - Probability = $\frac{100 - 58}{100} = 0.42$ **Final answers:** - Median $\approx 36.92$ - $Q_1 \approx 26.11$ - $Q_3 \approx 47.73$ - IQR $\approx 21.62$ - $P_{60} \approx 40.77$ - Probabilities: less than 25 = 0.23, between 20 and 45 = 0.55, greater than 40 = 0.42