4. A group of 100 adults recorded the amount of time, \(t\) minutes, they spent exercising each day. Their results are summarised in the table below.
| Time (t minutes) | Frequency (f) | Time midpoint (x) |
| \(0 \leqslant t < 15\) | 25 | 7.5 |
| \(15 \leqslant t < 30\) | 17 | 22.5 |
| \(30 \leqslant t < 60\) | 28 | 45 |
| \(60 \leqslant t < 120\) | 24 | 90 |
| \(120 \leqslant t \leqslant 240\) | 6 | 180 |
[You may use \(\sum \mathrm { f } x ^ { 2 } = 455\) 512.5]
A histogram is drawn to represent these data.
The bar representing the time \(0 \leqslant t < 15\) has width 0.5 cm and height 6 cm .
- Calculate the width and height of the bar representing a time of \(60 \leqslant t < 120\)
- Use linear interpolation to estimate the median time spent exercising by these adults each day.
- Find an estimate of the mean time spent exercising by these adults each day.
- Calculate an estimate for the standard deviation of these times.
- Describe, giving a reason, the skewness of these data.
Further analysis of the above data revealed that 18 of the 25 adults in the \(0 \leqslant t < 15\) group took no exercise each day.
- State, giving a reason, what effect, if any, this new information would have on your answers to
- the estimate of the median in part (b),
- the estimate of the mean in part (c),
- the estimate of the standard deviation in part (d).