1. **Problem Statement:** We have test scores of 30 students grouped into intervals with frequencies. We need to: i. Complete the cumulative frequency column. ii. Identify the modal class. iii. Determine the median class. iv. Estimate the mean score using the midpoint method. 2. **Given Data:** | Class Interval | Frequency (f) | | 40 - 49 | 2 | | 50 - 59 | 6 | | 60 - 69 | 10 | | 70 - 79 | 8 | | 80 - 89 | 3 | | 90 - 99 | 1 | 3. **Step i: Calculate Cumulative Frequency (CF):** Cumulative frequency is the running total of frequencies. - CF for 40-49 = 2 - CF for 50-59 = 2 + 6 = 8 - CF for 60-69 = 8 + 10 = 18 - CF for 70-79 = 18 + 8 = 26 - CF for 80-89 = 26 + 3 = 29 - CF for 90-99 = 29 + 1 = 30 4. **Step ii: Identify Modal Class:** Modal class is the class interval with the highest frequency. - Frequencies: 2, 6, 10, 8, 3, 1 - Highest frequency = 10 in class 60-69 5. **Step iii: Determine Median Class:** Median class contains the middle value. - Total frequency $N = 30$ - Median position = $\frac{N}{2} = 15$ - Find class where CF $\\geq 15$: - CF up to 50-59 = 8 (less than 15) - CF up to 60-69 = 18 (greater than 15) - Median class = 60-69 6. **Step iv: Estimate Mean Using Midpoint Method:** - Midpoint ($x$) of each class = $\frac{\text{Lower limit} + \text{Upper limit}}{2}$ - Calculate $f \times x$ for each class: - 40-49: midpoint = $\frac{40+49}{2} = 44.5$, $f=2$, $f\times x=2 \times 44.5=89$ - 50-59: midpoint = $54.5$, $f=6$, $f\times x=327$ - 60-69: midpoint = $64.5$, $f=10$, $f\times x=645$ - 70-79: midpoint = $74.5$, $f=8$, $f\times x=596$ - 80-89: midpoint = $84.5$, $f=3$, $f\times x=253.5$ - 90-99: midpoint = $94.5$, $f=1$, $f\times x=94.5$ - Sum of $f \times x = 89 + 327 + 645 + 596 + 253.5 + 94.5 = 2005$ - Mean $= \frac{\sum f x}{\sum f} = \frac{2005}{30}$ 7. **Simplify Mean:** $$\text{Mean} = \frac{2005}{30} = \frac{\cancel{2005}}{\cancel{30}}$$ Since 2005 and 30 have no common factors, simplify by division: $$\text{Mean} = 66.8333$$ **Final Answers:** - Cumulative frequencies: 2, 8, 18, 26, 29, 30 - Modal class: 60-69 - Median class: 60-69 - Estimated mean score: 66.83 (rounded to 2 decimal places)