1. **State the problem:** We have a frequency distribution of grades on a statistics exam and need to find (a) the relative frequency table and (b) the cumulative frequency table. 2. **Given data:** - Below 50: 1 - 50 - 59: 6 - 60 - 69: 8 - 70 - 79: 16 - 80 - 89: 15 - 90 - 100: 9 3. **Calculate total frequency:** $$\text{Total} = 1 + 6 + 8 + 16 + 15 + 9 = 55$$ 4. **(a) Relative frequency:** Relative frequency for each class is given by: $$\text{Relative frequency} = \frac{\text{Frequency}}{\text{Total frequency}}$$ Calculate each: - Below 50: $\frac{1}{55} \approx 0.0182$ - 50 - 59: $\frac{6}{55} \approx 0.1091$ - 60 - 69: $\frac{8}{55} \approx 0.1455$ - 70 - 79: $\frac{16}{55} \approx 0.2909$ - 80 - 89: $\frac{15}{55} \approx 0.2727$ - 90 - 100: $\frac{9}{55} \approx 0.1636$ 5. **(b) Cumulative frequency:** Cumulative frequency is the sum of frequencies up to that class: - Below 50: $1$ - 50 - 59: $1 + 6 = 7$ - 60 - 69: $7 + 8 = 15$ - 70 - 79: $15 + 16 = 31$ - 80 - 89: $31 + 15 = 46$ - 90 - 100: $46 + 9 = 55$ **Final answers:** (a) Relative Frequency Table: | Grade Range | Relative Frequency | |-------------|--------------------| | Below 50 | 0.0182 | | 50 - 59 | 0.1091 | | 60 - 69 | 0.1455 | | 70 - 79 | 0.2909 | | 80 - 89 | 0.2727 | | 90 - 100 | 0.1636 | (b) Cumulative Frequency Table: | Grade Range | Cumulative Frequency | |-------------|----------------------| | Below 50 | 1 | | 50 - 59 | 7 | | 60 - 69 | 15 | | 70 - 79 | 31 | | 80 - 89 | 46 | | 90 - 100 | 55 |