4 Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.
- Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.
| Length | | \(x ( \mathrm {~mm} )\) |
| |
| \(x \leqslant 298\) | 1 |
| \(298 < x \leqslant 300\) | 30 |
| \(300 < x \leqslant 301\) | 62 |
| \(301 < x \leqslant 302\) | 70 |
| \(302 < x \leqslant 304\) | 34 |
| \(x > 304\) | 3 |
The corresponding expected frequencies for the Normal distribution with parameters estimated by the sample statistics are
| Length | | \(x ( \mathrm {~mm} )\) |
| |
| \(x \leqslant 298\) | 1.49 |
| \(298 < x \leqslant 300\) | 37.85 |
| \(300 < x \leqslant 301\) | 55.62 |
| \(301 < x \leqslant 302\) | 58.32 |
| \(302 < x \leqslant 304\) | 44.62 |
| \(x > 304\) | 2.10 |
- It is thought that the lengths of tubes in another batch have an underlying distribution similar to that for the batch in part (i) but possibly with different location and dispersion parameters. A random sample of 10 tubes from this batch gives the following lengths (in mm ).
$$\begin{array} { l l l l l l l l l l }
301.3 & 301.4 & 299.6 & 302.2 & 300.3 & 303.2 & 302.6 & 301.8 & 300.9 & 300.8
\end{array}$$
(A) Discuss briefly whether it would be appropriate to use a \(t\) test to examine a hypothesis about the population mean length for this batch.
(B) Use a Wilcoxon test to examine at the \(10 \%\) significance level whether the population median length for this batch is 301 mm .