1. **Problem Statement:** Find the quartiles $Q_1$, $Q_2$, $Q_3$ and deciles $D_1$, $D_3$ for the wages of 65 employees at the R and B company given the grouped frequency distribution. 2. **Data and Frequencies:** Wages (USD) | Number of Employees (Frequency $f$) 250.00-259.99 | 8 260.00-269.99 | 10 270.00-279.99 | 16 280.00-289.99 | 14 290.00-299.99 | 10 300.00-309.99 | 5 310.00-319.99 | 2 Total employees $N = 65$. 3. **Calculate cumulative frequencies (CF):** - $CF_1 = 8$ - $CF_2 = 8 + 10 = 18$ - $CF_3 = 18 + 16 = 34$ - $CF_4 = 34 + 14 = 48$ - $CF_5 = 48 + 10 = 58$ - $CF_6 = 58 + 5 = 63$ - $CF_7 = 63 + 2 = 65$ 4. **Formulas for quartiles and deciles in grouped data:** $$Q_k = L + \left(\frac{\frac{kN}{4} - F}{f}\right) \times h$$ $$D_k = L + \left(\frac{\frac{kN}{10} - F}{f}\right) \times h$$ Where: - $L$ = lower class boundary of the class containing the quartile/decile - $N$ = total frequency - $F$ = cumulative frequency before the quartile/decile class - $f$ = frequency of the quartile/decile class - $h$ = class width 5. **Class width $h$:** $259.99 - 250.00 = 9.99 \approx 10$ 6. **Find positions:** - $Q_1$ position = $\frac{1 \times 65}{4} = 16.25$ - $Q_2$ position = $\frac{2 \times 65}{4} = 32.5$ - $Q_3$ position = $\frac{3 \times 65}{4} = 48.75$ - $D_1$ position = $\frac{1 \times 65}{10} = 6.5$ - $D_3$ position = $\frac{3 \times 65}{10} = 19.5$ 7. **Locate classes for each position:** - $D_1 = 6.5$ lies in first class (CF=8) - $Q_1 = 16.25$ lies in second class (CF=18) - $D_3 = 19.5$ lies in third class (CF=34) - $Q_2 = 32.5$ lies in third class (CF=34) - $Q_3 = 48.75$ lies in fourth class (CF=48) but since 48 < 48.75, it lies in fifth class (CF=58) 8. **Calculate each:** - For $D_1$ (class 1: 250-259.99): $L=250$, $F=0$, $f=8$, $h=10$ $$D_1 = 250 + \frac{6.5 - 0}{8} \times 10 = 250 + 8.125 = 258.125$$ - For $Q_1$ (class 2: 260-269.99): $L=260$, $F=8$, $f=10$, $h=10$ $$Q_1 = 260 + \frac{16.25 - 8}{10} \times 10 = 260 + 8.25 = 268.25$$ - For $D_3$ (class 3: 270-279.99): $L=270$, $F=18$, $f=16$, $h=10$ $$D_3 = 270 + \frac{19.5 - 18}{16} \times 10 = 270 + 0.9375 = 270.9375$$ - For $Q_2$ (class 3: 270-279.99): $$Q_2 = 270 + \frac{32.5 - 18}{16} \times 10 = 270 + 8.90625 = 278.90625$$ - For $Q_3$ (class 5: 290-299.99): $L=290$, $F=48$, $f=10$, $h=10$ $$Q_3 = 290 + \frac{48.75 - 48}{10} \times 10 = 290 + 0.75 = 290.75$$ 9. **Final answers:** - $Q_1 = 268.25$ - $Q_2 = 278.91$ - $Q_3 = 290.75$ - $D_1 = 258.13$ - $D_3 = 270.94$ These values represent the wage thresholds below which the respective percentages of employees fall.