1. **State the problem:** Calculate the range and standard deviation for daily website traffic data for the Politics and Sports sections, then compare variability. 2. **Identify the datasets:** - Politics: $\{25, 30, 22, 28, 27, 26, 29\}$ - Sports: $\{20, 35, 18, 32, 25, 22, 28\}$ 3. **Calculate the range:** Range is the difference between the maximum and minimum values. - Politics: $\max = 30$, $\min = 22$ so $\text{Range} = 30 - 22 = 8$ - Sports: $\max = 35$, $\min = 18$ so $\text{Range} = 35 - 18 = 17$ 4. **Calculate the mean for each section:** - Politics mean $\mu_P = \frac{25 + 30 + 22 + 28 + 27 + 26 + 29}{7} = \frac{187}{7} = 26.71$ - Sports mean $\mu_S = \frac{20 + 35 + 18 + 32 + 25 + 22 + 28}{7} = \frac{180}{7} = 25.71$ 5. **Calculate the variance and standard deviation:** For a dataset $x_1, x_2, ..., x_n$ with mean $\mu$, variance $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$ and standard deviation $\sigma = \sqrt{\sigma^2}$. - Politics variance: $$\frac{(25-26.71)^2 + (30-26.71)^2 + (22-26.71)^2 + (28-26.71)^2 + (27-26.71)^2 + (26-26.71)^2 + (29-26.71)^2}{7}$$ $$= \frac{( -1.71)^2 + 3.29^2 + (-4.71)^2 + 1.29^2 + 0.29^2 + (-0.71)^2 + 2.29^2}{7}$$ $$= \frac{2.92 + 10.82 + 22.18 + 1.66 + 0.08 + 0.50 + 5.24}{7} = \frac{43.40}{7} = 6.20$$ - Politics standard deviation: $$\sigma_P = \sqrt{6.20} = 2.49$$ - Sports variance: $$\frac{(20-25.71)^2 + (35-25.71)^2 + (18-25.71)^2 + (32-25.71)^2 + (25-25.71)^2 + (22-25.71)^2 + (28-25.71)^2}{7}$$ $$= \frac{(-5.71)^2 + 9.29^2 + (-7.71)^2 + 6.29^2 + (-0.71)^2 + (-3.71)^2 + 2.29^2}{7}$$ $$= \frac{32.60 + 86.32 + 59.49 + 39.56 + 0.50 + 13.77 + 5.24}{7} = \frac{237.48}{7} = 33.93$$ - Sports standard deviation: $$\sigma_S = \sqrt{33.93} = 5.83$$ 6. **Compare variability:** - Politics range = 8, standard deviation = 2.49 - Sports range = 17, standard deviation = 5.83 Sports section has greater variability in daily traffic. 7. **Impact on digital strategy:** Higher variability in Sports traffic means traffic is less predictable, possibly requiring adaptive content and marketing strategies to maintain engagement, while Politics traffic is more stable and consistent.