1. **Problem 1: Sample size using Slovin's formula with 2% margin of error** Slovin's formula for sample size $n$ is: $$n = \frac{N}{1 + N e^2}$$ where $N$ is the population size and $e$ is the margin of error. Given: - $N = 3000$ - $e = 0.02$ (2%) Calculate: $$n = \frac{3000}{1 + 3000 \times (0.02)^2} = \frac{3000}{1 + 3000 \times 0.0004} = \frac{3000}{1 + 1.2} = \frac{3000}{2.2} \approx 1363.64$$ So, the required sample size is approximately **1364**. 2. **Problem 2: Sample size with 95% confidence level** The margin of error $e$ is not directly given, but for 95% confidence level, the typical margin of error is 5% or 0.05 unless otherwise specified. Given: - $N = 5000$ - $e = 0.05$ Calculate: $$n = \frac{5000}{1 + 5000 \times (0.05)^2} = \frac{5000}{1 + 5000 \times 0.0025} = \frac{5000}{1 + 12.5} = \frac{5000}{13.5} \approx 370.37$$ So, the required sample size is approximately **370**. 3. **Problem 3: Complete the table for Scores of Students in Mathematics** | Score | Frequency ($f$) | Class Mark ($x$) | Lower Boundaries | Cumulative Frequency ($CF$) | |-------|-----------------|------------------|------------------|-----------------------------| | 11-15 | 1 | $\frac{11+15}{2} = 13$ | 10.5 | 1 | | 16-20 | 2 | $\frac{16+20}{2} = 18$ | 15.5 | 3 | | 21-25 | 5 | $\frac{21+25}{2} = 23$ | 20.5 | 8 | | 26-30 | 11 | $\frac{26+30}{2} = 28$ | 25.5 | 19 | | 31-35 | 12 | $\frac{31+35}{2} = 33$ | 30.5 | 31 | | 36-40 | 11 | $\frac{36+40}{2} = 38$ | 35.5 | 42 | | 41-45 | 5 | $\frac{41+45}{2} = 43$ | 40.5 | 47 | | 46-50 | 1 | $\frac{46+50}{2} = 48$ | 45.5 | 48 | Total frequency $n = 48$. 4. **Problem 4: Calculate Mean, Median, and Mode** **Mean**: $$\bar{x} = \frac{\sum f x}{n}$$ Calculate $\sum f x$: $$1\times13 + 2\times18 + 5\times23 + 11\times28 + 12\times33 + 11\times38 + 5\times43 + 1\times48 = 13 + 36 + 115 + 308 + 396 + 418 + 215 + 48 = 1549$$ Mean: $$\bar{x} = \frac{1549}{48} \approx 32.27$$ **Median**: Median class is where cumulative frequency reaches $\frac{n}{2} = 24$. From cumulative frequency: - Up to 19 (26-30) - Up to 31 (31-35) Median class is 31-35. Use median formula: $$\text{Median} = L + \left(\frac{\frac{n}{2} - F}{f_m}\right) \times c$$ where: - $L = 30.5$ (lower boundary of median class) - $F = 19$ (cumulative frequency before median class) - $f_m = 12$ (frequency of median class) - $c = 5$ (class width) Calculate: $$\text{Median} = 30.5 + \left(\frac{24 - 19}{12}\right) \times 5 = 30.5 + \frac{5}{12} \times 5 = 30.5 + 2.08 = 32.58$$ **Mode**: Mode class is the class with highest frequency, which is 31-35 with frequency 12. Use mode formula: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times c$$ where: - $L = 30.5$ (lower boundary of mode class) - $f_1 = 12$ (frequency of mode class) - $f_0 = 11$ (frequency before mode class, 26-30) - $f_2 = 11$ (frequency after mode class, 36-40) - $c = 5$ (class width) Calculate: $$\text{Mode} = 30.5 + \frac{12 - 11}{2 \times 12 - 11 - 11} \times 5 = 30.5 + \frac{1}{24 - 22} \times 5 = 30.5 + \frac{1}{2} \times 5 = 30.5 + 2.5 = 33.0$$ **Final answers:** - Sample size (2% margin): **1364** - Sample size (95% confidence): **370** - Mean score: **32.27** - Median score: **32.58** - Mode score: **33.0**