Standard deviation and quartiles of grouped puzzle times from a cumulative frequency curve, the sum of deviations from the mean and a conditional probability

In a timed challenge, \(120\) students each solve a logic puzzle. The time, \(t\) minutes, taken by each student is recorded and the results are grouped in the table.

1. Using mid-interval values, calculate an estimate of the standard deviation of the times.[3]
2. Copy and complete the cumulative frequency table below.[1]

   Time (\(t\) minutes)	\(t\le24\)	\(t\le28\)	\(t\le32\)	\(t\le36\)	\(t\le40\)	\(t\le44\)	\(t\le48\)
   Cumulative frequency	\(6\)		\(42\)				\(120\)

3. The cumulative frequency curve below represents the same \(120\) times.
   Use the curve to estimate each of the following, giving your answers to the nearest minute.
   1. The median time.
   2. The upper quartile of the times.[4]
4. The individual times are \(t_1,t_2,\ldots,t_{120}\) minutes and their mean is \(\bar{t}\) minutes. Find the value of \(\displaystyle\sum_{i=1}^{120}\left(t_i-\bar{t}\right)\).[2]
5. A student is chosen at random from the \(120\). It is known that this student took no more than \(40\) minutes. Find the probability that the student took more than \(28\) minutes.[4]