Given: Mean of 20 values \(\bar{x} = 60\) and standard deviation \(s = 4\).

(i) To find \(\Sigma x\) and \(\Sigma x^2\):

\(\Sigma x = 60 \times 20 = 1200\)

\(Using the formula for variance:\)

\(s^2 = \frac{\Sigma x^2}{n} - \left(\frac{\Sigma x}{n}\right)^2\)

\(4^2 = \frac{\Sigma x^2}{20} - 60^2\)

\(16 = \frac{\Sigma x^2}{20} - 3600\)

\(\frac{\Sigma x^2}{20} = 3616\)

\(\Sigma x^2 = 3616 \times 20 = 72320\)

(ii) For the combined 30 values:

\(\Sigma x = 1200 + 550 = 1750\)

\(\Sigma x^2 = 72320 + 40500 = 112800\)

Mean of 30 values:

\(\bar{x} = \frac{1750}{30} = 58.3\)

Variance of 30 values:

\(s^2 = \frac{112800}{30} - \left(\frac{1750}{30}\right)^2\)

\(s^2 = 3760 - 3412.89 = 357.11\)

Standard deviation:

\(s = \sqrt{357.11} \approx 18.9\)

To determine the most suitable measure of central tendency, we consider the mean, mode, and median:

• Mean: The mean is affected by extreme values. In this dataset, the value 768 is much larger than the others, skewing the mean upwards. Therefore, the mean is not suitable.
• Mode: The mode is the most frequently occurring value. In this dataset, all values are different, so there is no mode. Thus, the mode is not suitable.
• Median: The median is the middle value when the data is ordered. It is not affected by extreme values. Ordering the data: 180, 194, 227, 228, 229, 235, 238, 242, 246, 275, 311, 332, 768. The median is 238, which is a representative measure of central tendency for this dataset.

Therefore, the median is the most suitable measure of central tendency for these times.

First, calculate the new sums:

\(\Sigma x = 1923 + 166 + 172 + 182 = 2443\)

\(\Sigma x^2 = 337221 + 166^2 + 172^2 + 182^2 = 427485\)

Calculate the mean:

\(\text{Mean} = \frac{\Sigma x}{14} = \frac{2443}{14} = 174.5\)

Calculate the variance:

\(\text{Variance} = \frac{\Sigma x^2}{14} - \left(\frac{\Sigma x}{14}\right)^2 = \frac{427485}{14} - \left(\frac{2443}{14}\right)^2\)

\(= 30534.64 - 30480.25 = 54.39\)

Calculate the standard deviation:

\(\text{Standard deviation} = \sqrt{54.39} = 9.19\)