- Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.
| Length \(( \boldsymbol { l } )\) | Frequency \(( \boldsymbol { f } )\) |
| \(15 < l \leqslant 20\) | 19 |
| \(20 < l \leqslant 25\) | 35 |
| \(25 < l \leqslant 27\) | 16 |
| \(27 < l \leqslant 30\) | 15 |
| \(30 < l \leqslant 40\) | 3 |
A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .
- Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
- Use linear interpolation to estimate the median of \(l\)
The maximum length of log Gill can use in her stove is 26 cm .
Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove. - Show that \(x = 62\)
Gill randomly selects 4 logs from the bag.
- Using \(x = 62\), find the probability that all 4 logs will fit into her stove.
The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by
$$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
- Calculate
- the mean of \(W\)
- the variance of \(W\)