1. **State the problem:** Given a frequency distribution of distances traveled by tires until unusable, find: (i) Mode class (ii) Mean rolling distance using the mode class mean as the hypothetical mean or manual method (iii) Estimate the annual cost of tires for 10 four-wheeled vehicles driven 25,000 km/year with each tire costing 10000. 2. **Data from the table:** | Distance (in 1000 km) | Frequency (f) | |-----------------------|--------------| | 18 - 24 | 1 | | 24 - 30 | 6 | | 30 - 36 | 5 | | 36 - 42 | 26 | | 42 - 48 | 15 | | 48 - 54 | 7 | 3. **Find the mode class:** The mode class corresponds to the class interval with the highest frequency. Highest frequency = 26 for class 36 - 42. So, **mode class** = $36-42$ thousand km. 4. **Calculate the mean rolling distance:** We use the manual method with class marks and frequencies. - Calculate midpoints ($x_i$) for each class: $$x_i = \frac{lower + upper}{2}$$ | Interval | $x_i$ (midpoint) | Frequency $f_i$ | |----------|-----------------|----------------| | 18-24 | 21 | 1 | | 24-30 | 27 | 6 | | 30-36 | 33 | 5 | | 36-42 | 39 | 26 | | 42-48 | 45 | 15 | | 48-54 | 51 | 7 | - Take the mode class midpoint as the assumed mean, $a=39$. - Calculate $d_i = x_i - a$: $$d_i = x_i - 39$$ | $x_i$ | $d_i$ | $f_i$ | $f_i d_i$ | |-------|-------|-------|-----------| | 21 | -18 | 1 | -18 | | 27 | -12 | 6 | -72 | | 33 | -6 | 5 | -30 | | 39 | 0 | 26 | 0 | | 45 | 6 | 15 | 90 | | 51 | 12 | 7 | 84 | - Sum of frequencies: $\sum f_i = 1+6+5+26+15+7 = 60$ - Sum of $f_i d_i$: $-18 -72 -30 + 0 + 90 + 84 = 54$ - Apply mean formula: $$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} = 39 + \frac{54}{60} = 39 + 0.9 = 39.9$$ So, mean rolling distance is $39.9 \times 1000 = 39900$ km. 5. **Estimate the annual cost for the vehicles:** - Each vehicle has 4 tires, and there are 10 vehicles. - Total tires used per year = $10 \times 4 = 40$ - Each tire covers approximately 39900 km before unusable. - Annual distance per vehicle = 25000 km - Tires needed per vehicle per year: $$\frac{25000}{39900} \approx 0.6266$$ - Tires needed for 10 vehicles: $$0.6266 \times 40 = 25.06$$ tires (approx 26 tires) - Cost per tire = 10000 - Total estimated annual cost: $$26 \times 10000 = 260000$$ **Final answers:** (i) Mode class: 36 - 42 thousand km (ii) Mean rolling distance: 39900 km (iii) Estimated annual cost: 260000