1. **Problem Statement:** Calculate the harmonic mean and geometric mean for the given frequency distribution of monthly wages. 2. **Given Data:** Monthly wages intervals and number of workers: - 13-17: 2 workers - 18-22: 22 workers - 23-27: 19 workers - 28-32: 14 workers - 33-37: 3 workers - 38-42: 4 workers - 43-47: 6 workers - 48-52: 1 worker - 53-57: 1 worker 3. **Step 1: Find midpoints ($x_i$) of each class interval:** - $x_1 = \frac{13+17}{2} = 15$ - $x_2 = \frac{18+22}{2} = 20$ - $x_3 = \frac{23+27}{2} = 25$ - $x_4 = \frac{28+32}{2} = 30$ - $x_5 = \frac{33+37}{2} = 35$ - $x_6 = \frac{38+42}{2} = 40$ - $x_7 = \frac{43+47}{2} = 45$ - $x_8 = \frac{48+52}{2} = 50$ - $x_9 = \frac{53+57}{2} = 55$ 4. **Step 2: Calculate total number of workers ($N$):** $$N = 2 + 22 + 19 + 14 + 3 + 4 + 6 + 1 + 1 = 72$$ 5. **Step 3: Calculate Harmonic Mean (HM):** Formula: $$HM = \frac{N}{\sum \frac{f_i}{x_i}}$$ where $f_i$ is frequency and $x_i$ is midpoint. Calculate $\sum \frac{f_i}{x_i}$: $$\frac{2}{15} + \frac{22}{20} + \frac{19}{25} + \frac{14}{30} + \frac{3}{35} + \frac{4}{40} + \frac{6}{45} + \frac{1}{50} + \frac{1}{55}$$ Calculate each term: - $\frac{2}{15} = 0.1333$ - $\frac{22}{20} = 1.1$ - $\frac{19}{25} = 0.76$ - $\frac{14}{30} = 0.4667$ - $\frac{3}{35} = 0.0857$ - $\frac{4}{40} = 0.1$ - $\frac{6}{45} = 0.1333$ - $\frac{1}{50} = 0.02$ - $\frac{1}{55} = 0.0182$ Sum: $$0.1333 + 1.1 + 0.76 + 0.4667 + 0.0857 + 0.1 + 0.1333 + 0.02 + 0.0182 = 2.8172$$ Therefore, $$HM = \frac{72}{2.8172} \approx 25.54$$ 6. **Step 4: Calculate Geometric Mean (GM):** Formula: $$GM = \left( \prod x_i^{f_i} \right)^{\frac{1}{N}}$$ Calculate $\sum f_i \ln x_i$: Calculate $\ln x_i$ for each midpoint: - $\ln 15 = 2.7081$ - $\ln 20 = 2.9957$ - $\ln 25 = 3.2189$ - $\ln 30 = 3.4012$ - $\ln 35 = 3.5553$ - $\ln 40 = 3.6889$ - $\ln 45 = 3.8067$ - $\ln 50 = 3.9120$ - $\ln 55 = 4.0073$ Multiply by frequencies: - $2 \times 2.7081 = 5.4162$ - $22 \times 2.9957 = 65.9054$ - $19 \times 3.2189 = 61.1591$ - $14 \times 3.4012 = 47.6168$ - $3 \times 3.5553 = 10.6659$ - $4 \times 3.6889 = 14.7556$ - $6 \times 3.8067 = 22.8402$ - $1 \times 3.9120 = 3.9120$ - $1 \times 4.0073 = 4.0073$ Sum: $$5.4162 + 65.9054 + 61.1591 + 47.6168 + 10.6659 + 14.7556 + 22.8402 + 3.9120 + 4.0073 = 236.2785$$ Calculate GM: $$GM = e^{\frac{236.2785}{72}} = e^{3.2816} \approx 26.64$$ --- 7. **Problem (iii): Find missing frequency given arithmetic mean = 34** Given data (assumed class intervals and frequencies with one missing frequency $f$): - Classes: 13-17, 18-22, 23-27, 28-32, 33-37, 38-42, 43-47, 48-52, 53-57 - Frequencies: 2, 22, 19, 14, 3, 4, 6, 1, $f$ 8. **Step 1: Calculate midpoints ($x_i$) as before.** 9. **Step 2: Use formula for arithmetic mean (AM):** $$AM = \frac{\sum f_i x_i}{\sum f_i} = 34$$ Let total frequency be: $$N = 2 + 22 + 19 + 14 + 3 + 4 + 6 + 1 + f = 71 + f$$ Calculate $\sum f_i x_i$ without $f$ term: $$2 \times 15 + 22 \times 20 + 19 \times 25 + 14 \times 30 + 3 \times 35 + 4 \times 40 + 6 \times 45 + 1 \times 50 + f \times 55$$ Calculate known sum: - $2 \times 15 = 30$ - $22 \times 20 = 440$ - $19 \times 25 = 475$ - $14 \times 30 = 420$ - $3 \times 35 = 105$ - $4 \times 40 = 160$ - $6 \times 45 = 270$ - $1 \times 50 = 50$ Sum known terms: $$30 + 440 + 475 + 420 + 105 + 160 + 270 + 50 = 1950$$ 10. **Step 3: Set up equation for AM:** $$34 = \frac{1950 + 55f}{71 + f}$$ Multiply both sides: $$34(71 + f) = 1950 + 55f$$ $$2414 + 34f = 1950 + 55f$$ Rearranged: $$2414 - 1950 = 55f - 34f$$ $$464 = 21f$$ Solve for $f$: $$f = \frac{464}{21} \approx 22.10$$ Since frequency must be whole number, $f = 22$. --- **Final answers:** - Harmonic Mean $\approx 25.54$ - Geometric Mean $\approx 26.64$ - Missing frequency $f = 22$