(a) To find the standard deviation, we first calculate the variance using the formula:

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

Substitute the given values:

\(\text{Var} = \frac{14235}{50} - \left(\frac{700}{50}\right)^2\)

\(\text{Var} = 284.7 - 196 = 88.7\)

The standard deviation is the square root of the variance:

\(\text{sd} = \sqrt{88.7} \approx 9.42\)

(b) Given \(\Sigma x = 2865\), we use the equation:

\(\Sigma x - 50q = 700\)

Substitute \(\Sigma x = 2865\):

\(2865 - 50q = 700\)

Solve for \(q\):

\(50q = 2865 - 700\)

\(50q = 2165\)

\(q = \frac{2165}{50} = 43.3\)

(i) Given the mean of the n values of x is 24.25, we have:

\(\frac{136}{n} + 20 = 24.25\)

Solving for \(n\):

\(24.25n - 20n = 136\)

\(4.25n = 136\)

\(n = \frac{136}{4.25} = 32\)

(ii) Using the coded information for variance:

\(\text{Variance} = \frac{2888}{32} - \left(\frac{136}{32}\right)^2\)

\(= 72.1875 \approx 72.19\)

Using the uncoded information for variance:

\(\text{Variance} = \frac{\Sigma x^2}{32} - 24.25^2\)

Equating the two expressions for variance:

\(\frac{\Sigma x^2}{32} - 24.25^2 = 72.1875\)

\(\frac{\Sigma x^2}{32} = 72.1875 + 24.25^2\)

\(\Sigma x^2 = 32 \times (72.1875 + 24.25^2)\)

\(\Sigma x^2 = 21128\)

First, calculate \(\Sigma(x - 45)\):

\(\Sigma(x - 45) = \Sigma x - 20 \times 45 = 1218 - 20 \times 45 = 318\)

Next, use the variance formula to find \(\Sigma(x - 45)^2\):

The variance formula is \(\frac{\Sigma(x - 45)^2}{20} - \left(\frac{\Sigma(x - 45)}{20}\right)^2 = 4.2^2\)

Substitute \(\Sigma(x - 45) = 318\):

\(\frac{\Sigma(x - 45)^2}{20} - \left(\frac{318}{20}\right)^2 = 4.2^2\)

Calculate \(4.2^2 = 17.64\) and \(\left(\frac{318}{20}\right)^2 = 252.81\):

\(\frac{\Sigma(x - 45)^2}{20} = 17.64 + 252.81 = 270.45\)

\(\Sigma(x - 45)^2 = 270.45 \times 20 = 5409\)

We know that \(\Sigma(x - 10) = 27\) and the mean number of emails is 11.5. The mean is given by \(\frac{\Sigma x}{n} = 11.5\).

First, express \(\Sigma x\) in terms of \(n\):

\(\Sigma x = 11.5n\).

We also have:

\(\Sigma(x - 10) = \Sigma x - 10n = 27\).

Substitute \(\Sigma x = 11.5n\) into the equation:

\(11.5n - 10n = 27\).

Simplify the equation:

\(1.5n = 27\).

Solving for \(n\):

\(n = \frac{27}{1.5} = 18\).

(i) We know \(\Sigma (x - k) = 315\) and there are 30 buns, so the mean of \(x - k\) is \(\frac{315}{30} = 10.5\). Since the mean weight of the buns is 50.5 grams, we have:

\(50.5 - k = 10.5\)

Solving for \(k\), we get:

\(k = 50.5 - 10.5 = 40\)

(ii) The variance is given by:

\(\text{var} = \frac{\Sigma (x - k)^2}{30} = \frac{4022}{30} = 134.067\)

We also know the variance formula:

\(\text{var} = \frac{\Sigma x^2}{30} - (\text{mean})^2\)

Substituting the mean \(50.5\), we have:

\(\text{var} = 134.067 - 50.5^2 = 23.817\)

The standard deviation is the square root of the variance:

\(\text{sd} = \sqrt{23.817} = 4.88\)