(i) To find the mean, use the formula:

\(\text{Mean} = 45 + \frac{-148}{36} = 40.9\)

To find the variance, use:

\(\text{Var} = \frac{3089}{36} - \left(\frac{-148}{36}\right)^2 = 68.9\)

The standard deviation is:

\(\text{sd} = \sqrt{68.9} = 8.30\)

(ii) With the new data value of 29 added:

New \(\Sigma(x - 45) = -148 + (29 - 45) = -164\)

New \(\Sigma(x - 45)^2 = 3089 + (29 - 45)^2 = 3345\)

The new standard deviation is:

\(\text{New sd} = \sqrt{\frac{3345}{37} - \left(\frac{-164}{37}\right)^2} = 8.41\)

(i) To find the mean speed, use the formula for the mean: \(\bar{x} = 60 + \frac{\Sigma(x - 60)}{70}\). Given \(\Sigma(x - 60) = 245\), we have:

\(\bar{x} = 60 + \frac{245}{70} = 60 + 3.5 = 63.5\)

(ii) To find \(\Sigma(x - 50)\), use the relationship:

\(\Sigma(x - 50) = \Sigma x - 70 \times 50\)

We know \(\Sigma(x - 60) = 245\), so:

\(\Sigma x = 245 + 70 \times 60\)

\(\Sigma x = 245 + 4200 = 4445\)

Then:

\(\Sigma(x - 50) = 4445 - 3500 = 945\)

(iii) To calculate \(\Sigma(x - 50)^2\), use the variance formula:

\(\text{Variance} = \frac{\Sigma(x - 50)^2}{70} - \left(\frac{\Sigma(x - 50)}{70}\right)^2\)

Given the standard deviation is 10.6, the variance is \(10.6^2 = 112.36\).

Let \(\frac{\Sigma(x - 50)}{70} = 13.5\) (from part (ii)), then:

\(112.36 = \frac{\Sigma(x - 50)^2}{70} - 13.5^2\)

\(112.36 = \frac{\Sigma(x - 50)^2}{70} - 182.25\)

\(\frac{\Sigma(x - 50)^2}{70} = 112.36 + 182.25 = 294.61\)

\(\Sigma(x - 50)^2 = 294.61 \times 70 = 20623\)

We start with the given equations:

\(\Sigma(x - 200) = 446\)

\(\Sigma x = 6846\)

We can express \(\Sigma(x - 200)\) as:

\(\Sigma x - \Sigma 200 = 446\)

Since \(\Sigma 200 = 200n\), we substitute:

\(\Sigma x - 200n = 446\)

Substitute \(\Sigma x = 6846\) into the equation:

\(6846 - 200n = 446\)

Solve for \(n\):

\(6846 - 446 = 200n\)

\(6400 = 200n\)

\(n = \frac{6400}{200}\)

\(n = 32\)

(i) To find \(\Sigma(x - 50)\), use the formula:

\(\Sigma(x - 50) = \Sigma x - 16 \times 50 = 824 - 800 = 24\)

To find \(\Sigma(x - 50)^2\), use the variance formula:

\(\frac{\Sigma(x - 50)^2}{16} - \left(\frac{\Sigma(x - 50)}{16}\right)^2 = 6.5^2\)

\(\frac{\Sigma(x - 50)^2}{16} - \left(\frac{24}{16}\right)^2 = 42.25\)

\(\Sigma(x - 50)^2 = 712\)

(ii) With the new mark, the total sum becomes \(824 + 72 = 896\) and the number of students is 17.

New mean:

\(\text{New mean} = \frac{896}{17} = 52.7\)

New variance:

\(\text{New variance} = \frac{712 + 22^2}{17} - \left(\frac{24 + (72 - 50)}{17}\right)^2\)

\(= \frac{712 + 484}{17} - \left(\frac{46}{17}\right)^2\)

\(= \frac{1196}{17} - \left(2.7059\right)^2\)

\(= 70.3529 - 7.3248 = 63.0281\)

New standard deviation:

\(\text{New standard deviation} = \sqrt{63.0281} \approx 7.94\)

To find the mean \(\bar{t}\), use the formula:

\(\bar{t} = \frac{\Sigma t}{n}\)

where \(n = 50\) and \(\Sigma t = 910\).

\(\bar{t} = \frac{910}{50} = 18.2\)

To find the standard deviation, use the formula:

\(\text{standard deviation} = \sqrt{\frac{\Sigma (t - \bar{t})^2}{n}}\)

where \(\Sigma (t - \bar{t})^2 = 876\).

\(\text{standard deviation} = \sqrt{\frac{876}{50}} = 4.19\)