First, calculate \((x - 1760)\) for each value of \(x\):

1.6, -1.5, 2.3, 1.4, -0.6, -0.9, 2.5, 1.9, 2.4, 1.9, 2.8, 1.0

Calculate the mean of \((x - 1760)\):

\(\text{Mean} = \frac{1.6 + (-1.5) + 2.3 + 1.4 + (-0.6) + (-0.9) + 2.5 + 1.9 + 2.4 + 1.9 + 2.8 + 1.0}{12} = 1.23\)

Calculate the standard deviation of \((x - 1760)\):

\(\text{sd} = 1.39\)

Since \(x = (x - 1760) + 1760\), the mean of \(x\) is:

\(\text{Mean of } x = 1.23 + 1760 = 1761.23\)

The standard deviation remains the same, so:

\(\text{sd of } x = 1.39\)

(i) To find \(c\), use the equation for the mean: \(\frac{\Sigma x}{9} = 2205\). Therefore, \(\Sigma x = 2205 \times 9 = 19845\). Given \(\Sigma(x-c) = 1845\), we have \(\Sigma x - 9c = 1845\). Substituting \(\Sigma x = 19845\), we get:

\(19845 - 9c = 1845\)

\(9c = 19845 - 1845 = 18000\)

\(c = \frac{18000}{9} = 2000\)

(ii) The variance is given by \(\frac{\Sigma(x-c)^2}{9} - (\text{mean of } x)^2\). Substituting the given values:

Variance = \(\frac{477450}{9} - 2205^2\)

Variance = \(53050 - 2205^2\)

Variance = \(11025\)

(iii) With the new apartment, the mean becomes \(2120.50\) for 10 apartments. Therefore, the new total is:

\(2120.50 \times 10 = 21205\)

The rental price of the additional apartment is:

\(21205 - 19845 = 1360\)

Given that the mean of 10 values is 86.2, we have:

\(\frac{\Sigma x}{10} = 86.2\)

Therefore, \(\Sigma x = 86.2 \times 10 = 862\).

For part (ii), we know \(\Sigma(x-a) = 362\).

Expanding this, we have:

\(\Sigma x - 10a = 362\)

Substituting \(\Sigma x = 862\) from part (i):

\(862 - 10a = 362\)

Solving for \(a\):

\(10a = 862 - 362\)

\(10a = 500\)

\(a = 50\)

To find the standard deviation, we use the formula for the standard deviation of a sample:

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

Given \(\Sigma(t - 2.5)^2 = 96.1\) and \(n = 250\), we first find the variance:

\(\text{variance} = \frac{96.1}{250} - (0.3)^2\)

where \(0.3\) is the coded mean difference \(2.8 - 2.5\).

\(\text{variance} = \frac{96.1}{250} - 0.09\)

\(\text{variance} = 0.3844 - 0.09 = 0.2944\)

Now, calculate the standard deviation:

\(\text{sd} = \sqrt{0.2944} = 0.543\)

We start with the equation \(\Sigma (x - 100) = 216\).

This can be expanded to \(\Sigma x - 100n = 216\).

We are given \(\Sigma x = 2416\), so substitute this into the equation:

\(2416 - 100n = 216\).

Rearrange to solve for \(n\):

\(2416 - 216 = 100n\)

\(2200 = 100n\)

\(n = \frac{2200}{100} = 22\).