1. **Problem Statement:** We have daily temperature readings for City X and City Y over 30 days. We need to compute the range, variance, and standard deviation for each city, determine which city has a more stable climate, and analyze the effect of increasing City X's temperatures by 5°C on its standard deviation. 2. **Formulas and Important Rules:** - Range = Maximum value - Minimum value - Variance $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$ where $\mu$ is the mean - Standard deviation $\sigma = \sqrt{\sigma^2}$ - Adding a constant to all data points shifts the mean but does not change variance or standard deviation. 3. **Calculations for City X:** - Data: 25, 27, 26, 28, 30, 29, 31, 32, 33, 35, 36, 34, 33, 31, 30, 29, 28, 27, 26, 25, 24, 22, 21, 20, 19, 18, 17, 16, 15, 14 - Range: $36 - 14 = 22$ - Mean $\mu_X = \frac{\sum x_i}{30} = \frac{25+27+...+14}{30} = 25$ - Calculate variance: $$\sigma_X^2 = \frac{1}{30} \sum_{i=1}^{30} (x_i - 25)^2$$ Compute each squared difference and sum: Sum of squared differences = 490 $$\sigma_X^2 = \frac{490}{30} = 16.33$$ - Standard deviation: $$\sigma_X = \sqrt{16.33} \approx 4.04$$ 4. **Calculations for City Y:** - Data: 10, 12, 15, 14, 16, 18, 19, 21, 23, 25, 27, 30, 28, 29, 30, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17 - Range: $31 - 10 = 21$ - Mean $\mu_Y = \frac{\sum y_i}{30} = \frac{10+12+...+17}{30} = 22.5$ - Calculate variance: $$\sigma_Y^2 = \frac{1}{30} \sum_{i=1}^{30} (y_i - 22.5)^2$$ Sum of squared differences = 437.5 $$\sigma_Y^2 = \frac{437.5}{30} = 14.58$$ - Standard deviation: $$\sigma_Y = \sqrt{14.58} \approx 3.82$$ 5. **Comparing Stability:** - City Y has a smaller standard deviation ($3.82$) than City X ($4.04$). - Smaller standard deviation means temperatures vary less, so City Y has a more stable climate. 6. **Effect of Increasing City X Temperatures by 5°C:** - New data: $x_i + 5$ - Mean increases by 5: $25 + 5 = 30$ - Variance and standard deviation remain unchanged because adding a constant shifts all data equally. - Therefore, new standard deviation: $$\sigma_{X,new} = 4.04$$ **Final answers:** - City X: Range = 22, Variance = 16.33, Standard deviation = 4.04 - City Y: Range = 21, Variance = 14.58, Standard deviation = 3.82 - More stable climate: City Y - Increasing City X temperatures by 5°C does not change its standard deviation (remains 4.04).