1. **State the problem:** Given import data for 1984 from various Asian countries: $a = [297.2, 266.1, 820.1, 101.8, 431.9, 19.5, 8.4, 113.2, 3.7]$, find $a_3$, $a_4$ and compute the variance, standard deviation, coefficient of skewness, and coefficient of kurtosis. 2. **Find $a_3$ and $a_4$:** - Since indexing starts at 1, $a_3 = 820.1$. - Similarly, $a_4 = 101.8$. 3. **Calculate the mean \( \bar{a} \):** $$\bar{a} = \frac{1}{9} \sum_{i=1}^9 a_i = \frac{297.2 + 266.1 + 820.1 + 101.8 + 431.9 + 19.5 + 8.4 + 113.2 + 3.7}{9} = \frac{2061.9}{9} = 229.1$$ 4. **Calculate Variance \( \sigma^2 \):** - Compute squared deviations: \((297.2-229.1)^2=4624.81\) \((266.1-229.1)^2=1369\) \((820.1-229.1)^2=348102.4\) \((101.8-229.1)^2=16214.1\) \((431.9-229.1)^2=41498.2\) \((19.5-229.1)^2=43715.2\) \((8.4-229.1)^2=48700.1\) \((113.2-229.1)^2=13466.4\) \((3.7-229.1)^2=50616.2\) - Sum of squared deviations: $$4624.81 + 1369 + 348102.4 + 16214.1 + 41498.2 + 43715.2 + 48700.1 + 13466.4 + 50616.2 = 566206.4$$ - Variance: $$\sigma^2 = \frac{566206.4}{9-1} = \frac{566206.4}{8} = 70775.8$$ 5. **Calculate Standard Deviation \( \sigma \):** $$\sigma = \sqrt{70775.8} \approx 266.0$$ 6. **Calculate Coefficient of Skewness \( g_1 \):** - Compute normalized deviations cubed:\ \(\left(\frac{a_i - \bar{a}}{\sigma}\right)^3\) for each term: \(0.254^3 = 0.0164\), \(0.137^3 = 0.0026\), \(2.21^3 = 10.75\), \((-0.479)^3 = -0.110\), \(0.765^3 = 0.447\), \((-0.791)^3 = -0.495\), \((-0.826)^3 = -0.564\), \((-0.443)^3 = -0.087\), \((-0.848)^3 = -0.610\) - Sum: $$0.0164 + 0.0026 + 10.75 - 0.11 + 0.447 - 0.495 - 0.564 - 0.087 - 0.610 = 9.349$$ - Skewness: $$g_1 = \frac{9}{8 \times 7} \times 9.349 = \frac{9}{56} \times 9.349 \approx 1.503$$ 7. **Calculate Coefficient of Kurtosis (excess) \( g_2 \):** - Compute normalized deviations to the fourth power: \(0.254^4 = 0.0042\), \(0.137^4 = 0.0004\), \(2.21^4 = 23.76\), \((-0.479)^4 = 0.053\), \(0.765^4 = 0.342\), \((-0.791)^4 = 0.391\), \((-0.826)^4 = 0.465\), \((-0.443)^4 = 0.039\), \((-0.848)^4 = 0.517\) - Sum: $$0.0042 + 0.0004 + 23.76 + 0.053 + 0.342 + 0.391 + 0.465 + 0.039 + 0.517 = 25.57$$ - Compute constants: $$\frac{9 \times 10}{8 \times 7 \times 6} = \frac{90}{336} = 0.2679$$ $$3 \times \frac{8^2}{7 \times 6} = \frac{3 \times 64}{42} = 4.57$$ - Kurtosis: $$g_2 = 0.2679 \times 25.57 - 4.57 = 6.85 - 4.57 = 2.28$$ **Final answers:** - $a_3 = 820.1$ - $a_4 = 101.8$ - Variance $= 70775.8$ - Standard deviation $= 266.0$ - Coefficient of skewness $= 1.503$ - Coefficient of kurtosis $= 2.28$