13 At a certain factory Christmas tree decorations are packed in boxes of 10 .
The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged.
His results are displayed in Fig. 13.1.
\begin{table}[h]
| Number of damaged decorations | 0 | 1 | 2 | 3 | 4 | 5 or more |
| Number of boxes | 19 | 35 | 28 | 13 | 5 | 0 |
\captionsetup{labelformat=empty}
\caption{Fig. 13.1}
\end{table}
- Calculate
- the mean number of damaged decorations per box,
- the standard deviation of the number of damaged decorations per box.
It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\). - State suitable values for \(n\) and \(p\).
- Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place.
\begin{table}[h]
| Number of damaged decorations | 0 | 1 | 2 | 3 | 4 | 5 or more |
| Observed number of boxes | 19 | 35 | 28 | 13 | 5 | 0 |
| Expected number of boxes | | | | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 13.2}
- Explain whether the model is a good fit for these data.