1. **State the problem:** We have a sample data set: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55. We want to find the new standard deviation if every observation is multiplied by 3. 2. **Recall the formula and rule:** The standard deviation measures the spread of data around the mean. If each data point is multiplied by a constant $c$, the new standard deviation is $|c|$ times the original standard deviation. 3. **Calculate the original standard deviation:** - Find the mean: $$\bar{x} = \frac{5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55}{11} = \frac{330}{11} = 30$$ - Calculate each squared deviation and sum: $$\sum (x_i - \bar{x})^2 = (5-30)^2 + (10-30)^2 + \cdots + (55-30)^2$$ $$= 625 + 400 + 225 + 100 + 25 + 0 + 25 + 100 + 225 + 400 + 625 = 2750$$ - Calculate variance (sample variance with $n-1$): $$s^2 = \frac{2750}{11 - 1} = \frac{2750}{10} = 275$$ - Calculate standard deviation: $$s = \sqrt{275} \approx 16.58$$ 4. **Calculate new standard deviation:** - Multiply original standard deviation by 3: $$s_{new} = 3 \times 16.58 = 49.75$$ 5. **Answer:** The new standard deviation after multiplying each observation by 3 is **49.75**.