1. **Find the number of classes (k) for a dataset with 100 data points using Sturges’ rule.** Sturges' rule formula for number of classes is: $$k = 1 + \log_2(n)$$ where $n$ is the number of data points. For $n=100$: $$k = 1 + \log_2(100)$$ We know $\log_2(100) = \frac{\log_{10}(100)}{\log_{10}(2)} = \frac{2}{0.3010} \approx 6.644$$ So, $$k = 1 + 6.644 = 7.644$$ Since number of classes must be an integer, round to nearest whole number: $$k = 8$$ 2. **Construct the frequency distribution table for a dataset of 50 data points with less class intervals starting from 0, using Sturges’ rule.** First, find number of classes $k$: $$k = 1 + \log_2(50)$$ Calculate $\log_2(50)$: $$\log_2(50) = \frac{\log_{10}(50)}{\log_{10}(2)} = \frac{1.6990}{0.3010} \approx 5.644$$ So, $$k = 1 + 5.644 = 6.644 \approx 7$$ To have fewer classes, choose $k=5$ (less than 7). Assuming data range is from 0 to max value (not given), class width $w$ is: $$w = \frac{\text{max} - 0}{k}$$ Without max value, we cannot complete the table, but the classes start at 0 with width $w$. 3. **Find the class width when the number of classes is determined to be 6, and the range of the data is 18.** Class width formula: $$w = \frac{\text{Range}}{k}$$ Given $k=6$, Range = 18: $$w = \frac{18}{6} = 3$$ 4. **Apply Sturges’ rule to a dataset of 400 observations. Calculate the number of classes and then determine the class width if the range is 60.** Number of classes: $$k = 1 + \log_2(400)$$ Calculate $\log_2(400)$: $$\log_2(400) = \frac{\log_{10}(400)}{\log_{10}(2)} = \frac{2.6021}{0.3010} \approx 8.64$$ So, $$k = 1 + 8.64 = 9.64 \approx 10$$ Class width: $$w = \frac{60}{10} = 6$$