1. Problem 1: Frog lengths data: 7.3, 8.1, 7.8, 8.4, 7.9, 8.2, 7.6, 8.0, 7.7 (a) Range: The range is the difference between the maximum and minimum values. $$\text{Range} = \max - \min = 8.4 - 7.3 = 1.1$$ (b) Quartiles and IQR: Order the data: 7.3, 7.6, 7.7, 7.8, 7.9, 8.0, 8.1, 8.2, 8.4 Number of data points $n=9$. - Median is the 5th value: 7.9 - Lower half: 7.3, 7.6, 7.7, 7.8 - Upper half: 8.0, 8.1, 8.2, 8.4 First quartile $Q_1$ is median of lower half $(7.6 + 7.7)/2 = 7.65$ Third quartile $Q_3$ is median of upper half $(8.1 + 8.2)/2 = 8.15$ $IQR = Q_3 - Q_1 = 8.15 - 7.65 = 0.5$ (c) Sample variance $s^2$: Sample mean: $$\bar{x} = \frac{7.3 + 8.1 + 7.8 + 8.4 + 7.9 + 8.2 + 7.6 + 8.0 + 7.7}{9} = \frac{70.0}{9} \approx 7.778$$ Calculate squared deviations and sum: $$(7.3 - 7.778)^2 + (8.1 - 7.778)^2 + \dots + (7.7 - 7.778)^2 = 0.9643$$ Sample variance: $$s^2 = \frac{0.9643}{9-1} = \frac{0.9643}{8} = 0.1205$$ (d) Sample standard deviation $s$: $$s = \sqrt{0.1205} \approx 0.347$$ 2. Problem 2: Test scores: 42, 45, 39, 47, 44, 41, 46, 40, 43, 48, 44, 42 (a) Range: $$\max = 48, \min = 39$$ $$\text{Range} = 48 - 39 = 9$$ (b) Quartiles and IQR: Order data: 39, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48 $n=12$, median is average of 6th and 7th values: (43 + 44)/2 = 43.5 Lower half: 39, 40, 41, 42, 42, 43 Upper half: 44, 44, 45, 46, 47, 48 $Q_1$ median lower half: (41+42)/2=41.5 $Q_3$ median upper half: (45+46)/2=45.5 $IQR = 45.5 - 41.5 = 4$ (c) Sample variance: Mean: $$\bar{x} = \frac{42+45+39+47+44+41+46+40+43+48+44+42}{12} = \frac{521}{12} \approx 43.42$$ Sum squared deviations: $$\sum (x_i - \bar{x})^2 = 122.91$$ Sample variance: $$s^2 = \frac{122.91}{11} \approx 11.17$$ (d) Sample standard deviation: $$s = \sqrt{11.17} \approx 3.34$$ 3. Problem 3: Metal sheet thicknesses: 1.12, 1.09, 1.15, 1.11, 1.14, 1.10, 1.13, 1.08, 1.16, 1.07 (a) Range: $$\max = 1.16, \min = 1.07$$ $$\text{Range} = 1.16 - 1.07 = 0.09$$ (b) Quartiles and IQR: Order data: 1.07, 1.08, 1.09, 1.10, 1.11, 1.12, 1.13, 1.14, 1.15, 1.16 $n=10$, median is average of 5th and 6th: (1.11+1.12)/2=1.115 Lower half: 1.07, 1.08, 1.09, 1.10, 1.11 Upper half: 1.12, 1.13, 1.14, 1.15, 1.16 $Q_1$ median lower half: 1.09 $Q_3$ median upper half: 1.14 $IQR = 1.14 - 1.09 = 0.05$ (c) Sample variance: Mean: $$\bar{x} = \frac{1.12+1.09+1.15+1.11+1.14+1.10+1.13+1.08+1.16+1.07}{10} = 1.115$$ Sum squared deviations: $$\sum (x_i - \bar{x})^2 = 0.00925$$ Sample variance: $$s^2 = \frac{0.00925}{9} \approx 0.00103$$ (d) Sample standard deviation: $$s = \sqrt{0.00103} \approx 0.032$$