4. A quality control manager regularly samples 20 items from a production line and records the number of defective items \(x\). The results of 100 such samples are given in table 1 below.
\begin{table}[h]
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 or more |
| Frequency | 17 | 31 | 19 | 14 | 9 | 7 | 3 | 0 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
- Estimate the proportion of defective items from the production line.
The manager claimed that the number of defective items in a sample of 20 can be modelled by a binomial distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.
\begin{table}[h]
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 or more |
| 12.2 | 27.0 | \(r\) | 19.0 | \(s\) | 3.2 | 0.9 | 0.2 |
\captionsetup{labelformat=empty}
\caption{Table 2}
- Find the value of \(r\) and the value of \(s\) giving your answers to 1 decimal place.
- Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim.
- Explain what the analysis in part (c) tells the manager about the occurrence of defective items from this production line.