1 A blueberry farmer increased the amount of water sprayed over his berries to see what effect this had on their weight. The farmer weighed each of a random sample of 80 berries of the previous season's crop and each of a random sample of 100 berries of the new crop. The results are summarised in the following table, in which \(\bar { x }\) denotes the sample mean weight in grams, and \(s ^ { 2 }\) denotes an unbiased estimate of the relevant population variance.
| Sample size | \(\bar { x }\) | \(s ^ { 2 }\) |
| Previous season's crop \(( P )\) | 80 | 1.24 | 0.00356 |
| New crop \(( N )\) | 100 | 1.36 | 0.00340 |
- Calculate an estimate of \(\operatorname { Var } \left( \bar { X } _ { N } - \bar { X } _ { P } \right)\).
- Calculate a \(95 \%\) confidence interval for the difference in population mean weights.
- Give a reason why it is unnecessary to use a \(t\)-distribution in calculating the confidence interval.