9 A supermarket sells trays of peaches. Each tray contains 10 peaches. Often some of the peaches in a tray are rotten. The numbers of rotten peaches in a random sample of 150 trays are shown in Table 9.1.
\begin{table}[h]
| Number of rotten peaches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Frequency | 39 | 39 | 33 | 19 | 8 | 8 | 4 | 0 |
\captionsetup{labelformat=empty}
\caption{Table 9.1}
\end{table}
A manager at the supermarket thinks that the number of rotten peaches in a tray may be modelled by a binomial distribution.
- Use these data to estimate the value of the parameter \(p\) for the binomial model \(\mathrm { B } ( 10 , p )\).
The manager decides to carry out a goodness of fit test to investigate further. The screenshot in Fig. 9.2 shows part of a spreadsheet to assess the goodness of fit of the distribution \(\mathrm { B } ( 10 , p )\), using the value of \(p\) estimated from the data.
\begin{table}[h]
| - | A | B | C | D | E |
| 1 | Number of rotten peaches | Observed frequency | Binomial probability | Expected frequency | Chi-squared contribution |
| 2 | 0 | 39 | | | |
| 3 | 1 | 39 | | | 1.4229 |
| 4 | 2 | 33 | 0.2941 | 44.1167 | 2.8012 |
| 5 | 3 | 19 | 0.1629 | 24.4383 | 1.2102 |
| 6 | \(\geqslant 4\) | 20 | 0.0769 | 11.5311 | 6.2199 |
| 7 | | | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 9.2}
- Calculate the missing values in each of the following cells.
- C2
- D2
- E2
- Explain why the numbers for 4, 5, 6 and at least 7 rotten peaches have been combined into the single category of at least 4 rotten peaches, as shown in the spreadsheet.
- Carry out the test at the \(1 \%\) significance level.
- Using the values of the contributions, comment on the results of the test.