6. The owner of a small restaurant decides to change the menu. A trade magazine claims that \(40 \%\) of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.

1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners \(X\) who choose organic foods.
2. Find \(\mathrm { P } ( 5 < X < 15 )\).
3. Find the mean and standard deviation of \(X\). The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.
4. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly.
   (5)