1. **Problem Statement:** We have two sets of data: house prices from Neighborhood A and B, and patient waiting times from Emergency Room and General Consultation. We need to calculate variance, standard deviation, mean, and range, then compare variability and answer related questions. --- ### Part 1: Neighborhood House Prices 2. **Formulas:** - Mean: $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$ - Variance: $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$ - Standard deviation: $s = \sqrt{s^2}$ 3. **Calculate for Neighborhood A:** - Data: 10.5, 12.3, 11.8, 9.7, 13.0, 15.2, 14.0, 12.1, 10.8, 11.3 - Mean: $$\bar{x}_A = \frac{10.5 + 12.3 + 11.8 + 9.7 + 13.0 + 15.2 + 14.0 + 12.1 + 10.8 + 11.3}{10} = \frac{120.7}{10} = 12.07$$ - Calculate squared deviations: $$\sum (x_i - \bar{x}_A)^2 = (10.5-12.07)^2 + (12.3-12.07)^2 + \cdots + (11.3-12.07)^2 = 19.001$$ - Variance: $$s_A^2 = \frac{19.001}{10-1} = \frac{19.001}{9} = 2.111$$ - Standard deviation: $$s_A = \sqrt{2.111} = 1.453$$ 4. **Calculate for Neighborhood B:** - Data: 8.5, 9.0, 9.3, 9.7, 10.0, 10.2, 10.5, 10.8, 11.0, 11.5 - Mean: $$\bar{x}_B = \frac{8.5 + 9.0 + 9.3 + 9.7 + 10.0 + 10.2 + 10.5 + 10.8 + 11.0 + 11.5}{10} = \frac{100.5}{10} = 10.05$$ - Squared deviations sum: $$\sum (x_i - \bar{x}_B)^2 = (8.5-10.05)^2 + (9.0-10.05)^2 + \cdots + (11.5-10.05)^2 = 8.025$$ - Variance: $$s_B^2 = \frac{8.025}{9} = 0.892$$ - Standard deviation: $$s_B = \sqrt{0.892} = 0.945$$ 5. **Comparison:** - Neighborhood A variance = 2.111, std dev = 1.453 - Neighborhood B variance = 0.892, std dev = 0.945 - **Neighborhood A has greater disparity in house prices.** --- ### Part 2: Hospital Waiting Times 6. **Calculate mean, standard deviation, and range for Emergency Room:** - Data: 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 - Mean: $$\bar{x}_{ER} = \frac{15 + 20 + \cdots + 60}{10} = \frac{375}{10} = 37.5$$ - Squared deviations sum: $$\sum (x_i - 37.5)^2 = 2062.5$$ - Variance: $$s_{ER}^2 = \frac{2062.5}{9} = 229.167$$ - Standard deviation: $$s_{ER} = \sqrt{229.167} = 15.14$$ - Range: $$60 - 15 = 45$$ 7. **Calculate mean, standard deviation, and range for General Consultation:** - Data: 10, 12, 14, 15, 16, 18, 20, 22, 24, 26 - Mean: $$\bar{x}_{GC} = \frac{10 + 12 + \cdots + 26}{10} = \frac{177}{10} = 17.7$$ - Squared deviations sum: $$\sum (x_i - 17.7)^2 = 280.1$$ - Variance: $$s_{GC}^2 = \frac{280.1}{9} = 31.12$$ - Standard deviation: $$s_{GC} = \sqrt{31.12} = 5.58$$ - Range: $$26 - 10 = 16$$ 8. **Comparison:** - Emergency Room std dev = 15.14, range = 45 - General Consultation std dev = 5.58, range = 16 - **Emergency Room has more variability in waiting times.** 9. **Patient choice:** - General Consultation has lower mean waiting time (17.7 min vs 37.5 min) - **Patient should go to General Consultation to minimize waiting time.** --- ### Part 3: Industry Standard Deviation Check 10. The industry standard requires maximum deviation ±0.5 cm from the mean. - This applies to machines, but no machine data was provided in the problem. - **No machine data given, so cannot determine if any machine fails this requirement.**