Some Useful Statistics links for a Level and Above

http://www.statslab.cam.ac.uk/Dept/People/djsteaching/teaching15.html

http://understandinguncertainty.org/

https://www.causeweb.org/wiki/chance/index.php/Main_Page

Everyone in the UK is the same age!!

Proof – which step does this proof go wrong in?

Statement S(n): In any group of n people, everyone in that group has the same age. E.g. All the people in the UK have the same age.

1. In any group that consists of just one person, everybody in the group has the same age, because after all there is only one person!
2. Therefore, statement S(1) is true.
3. The next stage in the induction argument is to prove that, whenever S(n) is true for one number (say n=k), it is also true for the next number (that is, n = k+1).
4. We can do this by (1) assuming that, in every group of k people, everyone has the same age; then (2) deducing from it that, in every group of k+1 people, everyone has the same age.
5. Let G be an arbitrary group of k+1 people; we just need to show that every member of G has the same age.
6. To do this, we just need to show that, if P and Q are any members of G, then they have the same age.
7. Consider everybody in G except P. These people form a group of k people, so they must all have the same age (since we are assuming that, in any group of k people, everyone has the same age).
8. Consider everybody in G except Q. Again, they form a group of k people, so they must all have the same age.
9. Let R be someone else in G other than P or Q.
10. Since Q and R each belong to the group considered in step 7, they are the same age.
11. Since P and R each belong to the group considered in step 8, they are the same age.
12. Since Q and R are the same age, and P and R are the same age, it follows that P and Q are the same age.
13. We have now seen that, if we consider any two people P and Q in G, they have the same age. It follows that everyone in G has the same age.
14. The proof is now complete: we have shown that the statement is true for n=1, and we have shown that whenever it is true for n=k it is also true for n=k+1, so by induction it is true for all n.

L6 Homework for Monday 9th Feb

ANSWERS

3rd Year Hwk Due Tue 10th Feb 15

To be done in front of books neatly. Show all working out. Marks: 21

ANSWERS

1. Solve \[ \begin{align*} 3&x+4y=19 \\ -5&x+6y=59 \end{align*} \]

   [3 marks]

2. A jacket is reduced by £45 in a 15% OFF sale. What was the price of the jacket in the sale?

   [3 marks]

3. A ball bounces to 90% of the previous height. If it was dropped from 5m, how high will it reach after the third bounce?

   [3 marks]

4. Give a quick explanation of why the sine ratio is never more than 1.

   [2 marks]

5. In a test there are 50 questions. For each correct answer I score 7 points and for each incorrect answer I lose 3 points. If my total score for the test was 260 points, set up simultaneous equations to find x (the number of questions I got correct) and y (the number of questions I got wrong).

   [4 marks]

6. Find x to 2decimal places
   

   [3 marks]

7. Find the angle x to 1 decimal place.
   

   [3 marks]