8 A soft drinks factory produces lemonade which is sold in packs of 6 bottles. As part of the factory's quality control, random samples of 75 packs are examined at regular intervals. The number of underfilled bottles in a pack of 6 bottles is denoted by the random variable \(X\). The results of one quality control check are shown in the following table.
| Number of underfilled bottles | 0 | 1 | 2 | 3 |
| Number of packs | 44 | 20 | 8 | 3 |
A researcher assumes that \(X \sim \mathrm {~B} ( 3 , p )\).
- By finding the sample mean, show that an estimate of \(p\) is 0.2 .
- Show that, at the \(5 \%\) significance level, there is evidence that this binomial distribution does not fit the data.
- Another researcher suggests that the goodness of fit test should be for \(\mathrm { B } ( 6 , p )\). She finds that the corresponding value of \(\chi ^ { 2 }\) is 2.74 , correct to 3 significant figures. Given that the number of degrees of freedom is the same as in part (ii), state the conclusion of the test at the same significance level.