3 At an agricultural research station, trials are being carried out to compare a standard variety of tomato with one that has been genetically modified (GM). The trials are concerned with the mean weight of the tomatoes and also with the aesthetic appearance of the tomatoes.

1. 1. Tomatoes of the standard and GM varieties are grown under similar conditions. The tomatoes are weighed and the data are summarised as follows.

      Variety	Sample size	Sum of weights \(( \mathrm { g } )\)	Sum of squares of weights \(\left( \mathrm { g } ^ { 2 } \right)\)
      Standard	30	3218.3	349257
      GM	26	2954.1	338691

      Carry out a test, using the Normal distribution, to investigate whether there is evidence, at the 5\% level of significance, that the two varieties of tomato differ in mean weight. State one assumption required for this test to be valid.
   2. The data in part (i) could have been used to carry out a test for the equality of means based on the \(t\) distribution. State two additional assumptions required for this test to be valid. Discuss briefly which test would be preferable in this case.
2. In order to judge whether, on the whole, GM tomatoes have a better aesthetic appearance than standard tomatoes, a trial is carried out as follows. 10 of each variety are chosen and consumer panel is asked to arrange the 20 tomatoes in order according to their appearance.
   1. State two important features of the way in which this trial should be designed. Comment briefly on how reliable the evidence from the trial is likely to be.
   2. The order in which the consumer panel arranges the tomatoes is as follows. The tomato with best appearance is listed first. \(G\) and \(S\) denote GM and standard tomatoes respectively. $$\begin{array} { c c c c c c c c c c c c c c c c c c c c } G & G & G & S & G & G & G & S & G & S & S & S & G & G & S & G & S & S & S & S \end{array}$$ Carry out an appropriate test at the \(1 \%\) level of significance.