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:** | Test Score Range | Frequency (f) | |------------------|--------------| | 40 - 49 | 2 | | 50 - 59 | 6 | | 60 - 69 | 10 | | 70 - 79 | 8 | | 80 - 89 | 3 | | 90 - 99 | 1 | 3. **Step i: Cumulative Frequency (CF)** Cumulative frequency is the running total of frequencies. $$\text{CF}_1 = 2$$ $$\text{CF}_2 = 2 + 6 = 8$$ $$\text{CF}_3 = 8 + 10 = 18$$ $$\text{CF}_4 = 18 + 8 = 26$$ $$\text{CF}_5 = 26 + 3 = 29$$ $$\text{CF}_6 = 29 + 1 = 30$$ 4. **Step ii: Modal Class** The modal class is the class interval with the highest frequency. Frequencies: 2, 6, 10, 8, 3, 1 The highest frequency is 10, so the modal class is **60 - 69**. 5. **Step iii: Median Class** Median class is the class containing the median position. Total frequency $N = 30$ Median position = $\frac{N}{2} = 15$ Locate the class where cumulative frequency just exceeds or equals 15: CFs: 2, 8, 18, 26, 29, 30 Since 18 is the first CF $\geq 15$, median class is **60 - 69**. 6. **Step iv: Estimate Mean Using Midpoint Method** - Find midpoints ($x$) of each class: - 40-49: $\frac{40+49}{2} = 44.5$ - 50-59: $\frac{50+59}{2} = 54.5$ - 60-69: $\frac{60+69}{2} = 64.5$ - 70-79: $\frac{70+79}{2} = 74.5$ - 80-89: $\frac{80+89}{2} = 84.5$ - 90-99: $\frac{90+99}{2} = 94.5$ - Multiply midpoints by frequencies ($f \times x$): - $2 \times 44.5 = 89$ - $6 \times 54.5 = 327$ - $10 \times 64.5 = 645$ - $8 \times 74.5 = 596$ - $3 \times 84.5 = 253.5$ - $1 \times 94.5 = 94.5$ - Sum of $f \times x$: $$89 + 327 + 645 + 596 + 253.5 + 94.5 = 2005$$ - Mean estimate: $$\text{Mean} = \frac{\sum f x}{\sum f} = \frac{2005}{30}$$ - Simplify fraction: $$\frac{2005}{30} = \frac{\cancel{2005}}{\cancel{30}}$$ (no common factors to cancel, so keep as is) - Calculate decimal: $$\approx 66.83$$ **Final answer:** Estimated mean score is approximately **66.83**. --- **Summary:** - Cumulative frequencies: 2, 8, 18, 26, 29, 30 - Modal class: 60 - 69 - Median class: 60 - 69 - Estimated mean: 66.83 **q_count:** 1