To find the mean, we use the formula:

\(\bar{x} = 80 - \frac{\Sigma (x - 80)}{30}\)

\(\bar{x} = 80 - \frac{-147}{30} = 80 + 4.9 = 75.1\)

To find the standard deviation, we use the formula:

\(\text{sd} = \sqrt{\frac{\Sigma (x - 80)^2}{30} - \left(\frac{\Sigma (x - 80)}{30}\right)^2}\)

\(\text{sd} = \sqrt{\frac{952}{30} - \left(\frac{-147}{30}\right)^2}\)

\(\text{sd} = \sqrt{31.7333 - 4.9^2} = \sqrt{7.72} \approx 2.78\)

First, calculate \(\Sigma x\) using the given \(\Sigma(x - 20) = 35\):

\(\Sigma x - 50 \times 20 = 35\)

\(\Sigma x = 35 + 1000 = 1035\)

Now, find the mean \(\bar{x}\):

\(\bar{x} = \frac{\Sigma x}{50} = \frac{1035}{50} = 20.7\)

Use the formula for variance:

\(\text{Variance} = \frac{\Sigma x^2}{50} - \left(\frac{\Sigma x}{50}\right)^2\)

Substitute the values:

\(\text{Variance} = \frac{25036}{50} - \left(\frac{1035}{50}\right)^2\)

\(\text{Variance} = 500.72 - (20.7)^2\)

\(\text{Variance} = 500.72 - 428.49\)

\(\text{Variance} = 72.23\)

(i) Calculate \(\Sigma(x - 62)\):

\((62.7 - 62) + (59.6 - 62) + (64.2 - 62) + (61.5 - 62) + (68.3 - 62) + (66.9 - 62) + (62.0 - 62) + (62.3 - 62) = 0.7 - 2.4 + 2.2 - 0.5 + 6.3 + 4.9 + 0 + 0.3 = 11.5\)

(ii) Calculate \(\Sigma(x - 62)^2\):

\((0.7)^2 + (-2.4)^2 + (2.2)^2 + (-0.5)^2 + (6.3)^2 + (4.9)^2 + (0)^2 + (0.3)^2 = 0.49 + 5.76 + 4.84 + 0.25 + 39.69 + 24.01 + 0 + 0.09 = 75.13\)

(iii) Calculate the mean and variance:

Mean \(= \frac{62.7 + 59.6 + 64.2 + 61.5 + 68.3 + 66.9 + 62.0 + 62.3}{8} = \frac{507.5}{8} = 63.4375\)

Variance \(= \frac{75.13}{8} - \left(\frac{11.5}{8}\right)^2 = 9.39125 - 2.06125 = 7.32\)

First, calculate \(\Sigma x\) using the mean:

\(\text{Mean} = \frac{\Sigma x}{18} = \frac{58}{9}\)

\(\Sigma x = 18 \times \frac{58}{9} = 116\)

Now, find \(\Sigma (x - 5)\):

\(\Sigma (x - 5) = \Sigma x - 18 \times 5 = 116 - 90 = 26\)

Next, calculate \(\Sigma (x - 5)^2\) using the formula:

\(\Sigma (x - 5)^2 = \Sigma x^2 - 2 \times 5 \times \Sigma x + 18 \times 5^2\)

\(\Sigma (x - 5)^2 = 967 - 2 \times 5 \times 116 + 18 \times 25\)

\(\Sigma (x - 5)^2 = 967 - 1160 + 450 = 257\)

First, find the mean of the speeds:

\(\bar{x} = 50 + \frac{81.4}{22} = 53.7\)

Next, calculate the variance using the formula:

\(\text{Var} = \frac{\Sigma(x - 50)^2}{22} - (\bar{x} - 50)^2\)

\(\text{Var} = \frac{671}{22} - 3.7^2 = 16.81\)

To find \(\Sigma x^2\), use the expanded equation:

\(\Sigma x^2 = \Sigma(x - 50)^2 + 2 \times 50 \times \Sigma(x - 50) + 22 \times 50^2\)

\(\Sigma x^2 = 671 + 2 \times 50 \times 81.4 + 22 \times 2500\)

\(\Sigma x^2 = 671 + 8140 + 55000 = 63811\)