1. **Problem Statement:** We have a frequency distribution of lottery sales over 30 days with class intervals and number of days as frequencies. | No of lotteries | 150-250 | 250-350 | 350-450 | 450-550 | 550-650 | 650-750 | 750-850 | | No of days | 3 | 2 | 5 | 9 | 3 | 6 | 2 | We need to find: (i) Modal class (ii) Mean number of lotteries sold per day (iii) Total commission received by the seller per day (iv) Show that the distributor earns more than 26000 rupees in a month 2. **Step 1: Find the modal class** The modal class is the class interval with the highest frequency (number of days). Frequencies: 3, 2, 5, 9, 3, 6, 2 Maximum frequency = 9 corresponding to class 450-550 **Modal class = 450-550** 3. **Step 2: Find the mean number of lotteries sold per day** We use the assumed mean method. - Take the mid value of the modal class as assumed mean $A$. Mid value of 450-550 = $\frac{450 + 550}{2} = 500$ - Calculate mid values ($x_i$) of all classes: 150-250: $\frac{150+250}{2} = 200$ 250-350: $300$ 350-450: $400$ 450-550: $500$ 550-650: $600$ 650-750: $700$ 750-850: $800$ - Calculate deviations $d_i = x_i - A$: 200 - 500 = -300 300 - 500 = -200 400 - 500 = -100 500 - 500 = 0 600 - 500 = 100 700 - 500 = 200 800 - 500 = 300 - Frequencies $f_i$: 3, 2, 5, 9, 3, 6, 2 - Calculate $f_i d_i$: 3*(-300) = -900 2*(-200) = -400 5*(-100) = -500 9*0 = 0 3*100 = 300 6*200 = 1200 2*300 = 600 - Sum of frequencies $\sum f_i = 3+2+5+9+3+6+2 = 30$ - Sum of $f_i d_i = -900 - 400 - 500 + 0 + 300 + 1200 + 600 = 300$ - Mean $\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} = 500 + \frac{300}{30} = 500 + 10 = 510$ - Rounded to nearest whole number: **510** 4. **Step 3: Calculate total commission received by the seller per day** - Commission per lottery = 3 rupees - Mean lotteries sold per day = 510 - Total commission per day = $3 \times 510 = 1530$ 5. **Step 4: Show distributor earns more than 26000 rupees in a month** - Selling price per lottery = 20 rupees - Total sales per day = $20 \times 510 = 10200$ - Commission paid to seller per day = 1530 rupees - Remaining amount per day = $10200 - 1530 = 8670$ - Distributor receives 10% of remaining amount: $0.10 \times 8670 = 867$ - Distributor's monthly earning = $867 \times 30 = 26010$ - Since 26010 > 26000, distributor earns more than 26000 rupees in a month. **Final answers:** (i) Modal class = 450-550 (ii) Mean number of lotteries sold per day = 510 (iii) Total commission per day = 1530 (iv) Distributor earns 26010 rupees in a month, which is more than 26000