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Saturday, February 18, 2006 

Scribe

Our first project was assigned, and it's due on Monday, February 27th. We must hand it in at the beginning of class that day. We also got a sheet which must be handed in with the project. So if you missed class, see Mr. K on Monday to get it. There must be two signitures on it, one your own, and the other one of your parent's or a guardian. Mr. K will not accept the project otherwise.

The assignment conditions are that we're allowed to use any books, notes, a calculator, a graphing calculators, or a computer to guide us. Other than that, we're not to get any other kind of help. That said, we're allowed to work in groups of two, but no more than that. We may also ask Mr. K. We were advised to get the actual math part of the project completed over the weekend. That's 50% of the grade. We're allowed to see Mr. K on Monday to see that our math is correct. He believes that the minimum grade we should receive on this project is 88%.

As mentioned, the actual math is worth 50%. We may either get A (the student has correctly solved the problem, with few or no mechanical errors), B (the student has used the correct method, but has made a minor error in the use of mathematical techniques) , C (the student has found a partial solution to the problem), or D (the student has had little or no sucess in solving the problem) within that aspect of the project.

Communication is something else we're being graded on, and it's worth 30% of the project. In this part of the project, we may be given either E, H, S, or U. E for exceptional, H for highly satisfactory, S for satisfactory, and U for unsatisfactory.

Finally, we'll be graded on Presentation. That's worth 20%. We may be given E, H, S, or U.

Okay, so now you're ready to know.. dun dun dun.. the PROJECT.


Matrices - A Growing Community
Task A.
A powerful use of matrices is to describe and explore situations in which one state depends on a previous state. A transition matrix can neatly describe the transition from one state to the next.
Consider the growth of populations. Suppose that the people in a (fictitous) country can be divided into three groups; children (aged up to 12 years), youth (aged from 12 to 19), and adults (aged over 19). Suppose further that the following is known about the population of each.
  • 8% of the children become youths (because they get a year older)
  • 0.5% of the children die
  • 2% of the adults die
  • 12.5% of the youths become adults (because they get a year older)
  • 1% of the youths die
  • The number of babies born is 4% of the adult population

The number of people in each category in a particular year can be represented as 3 x 1. Matrix A= [c]
[y]
[a]

a) Formulate equations to represent the number of people the following year.
b) Summarise these relationships into a 3 x 3 transition matrix B

Enter the matrix B into your graphing calculator. To use the matrices to model the population change, you need to know what the initial population is. Suppose that c=250 000, y=150 000, a=500 000. Enter this information into the calculator as Matrix A, a 3x1 *column matrix.

*We're used to row matrices, keep that in mind while doing the multiplication (how we multiply)

With the above assumptions, the calculator will allow you to predict the next year's population as BA. It is best to start by evaluating BA; then find Mat B x Mat Ans.

c) What is the population predicted to be after one year? Describe the changes.
d) Repeat the command Mat B x Mat Ans and record the population for the next 14 years.
e) Comment on the distribution of the population over the 15 years.
f) Use a spreedsheet to table each category of the population over 50 years.
g) Graph the categories over this time.
h) Comment on the result and predict the future for the community.

TASK B 'Population Variations'
Investigate the long-term effect of the adult death rate rising from 2% per year to 4% per year.

Wow.. I wish I could say that I was almost done explaining what we did in class, but the fun's just beginning.

We had the pre-test. Then we got into our groups to correct. I suppose I could post the answers.. :P

1) Three stores, A, B, and C, were surveyed to determine costs of the same weight and brand of bread, flour, and sugar. The amounts to be purchased and the prices charged are shown in the matrices below. If all three items are purchased at the same store, what is the lowest total price?

Amounts ABC
bread flour sugar = [3 1 2]
Prices A B C
Bread [1.09 1.15 1.03 ]
Flour [6.15 5.95 6.45 ]
Sugar [4.32 4.30 4.10]

a) $11.40 b) 11.58 c) $17.74 d) $18.00 -- The answer is c. Just multiply Amounts matrix x Prices matrix.

2) A network matrix, N, shows direct routes between cities. Which matrix calculation determines the number of routes with exactly k stopovers between two cities in the network?

a) Nk -1 b) Nk c) Nk +1 d) Nk +2 -- The answer is C. I think a lot of people in the class got this one wrong. I honestly can't remember what the explanation was, my bad.

3) Determine (3Q)2 =
[-1 1]
[1 2] All it is, is using scalar multiplication 3 x the matrix, then squaring it. The result is

[18 9]
[9 45]

4) Solve for x:
[1 0] [x ]
[x 6] [0 3][-2] = 0 . First, do x times 1, then x times 0. That makes x. Then it's 6 times o, and 6 times 3. That makes 18. Hence, [x 18]. x times x makes x2. 18 x -2 makes -36. Now, because we have to make this equal zero, x can be -6 or 6.

5) A washing powder is sold in regular and large boxes. Market research shows that 40% of the buyers of the regular size switch to the large size for their next purchase, and 18% of the buyers of the large size switch to the regular size for their next purchase. The original market share was $70 for the regular size and 30% for the large size.

a) What is the market share for each size in the next round of purchases?
r l r l r l
[.70 .30] x r [.60 .40] = [.474 .526]
l [.18 .82]
This means that 47.4% is the next market share for regular powder, and 52.6 is the market share for large powder.

b) In the long term, what is the market share for each size of this washing powder?

For this, just do the same thing, except to the power of 50. :)

Wow, so close to being done.. yet so far. :P

The final thing we did (that I remember) was a group assignment regarding probability (next unit we're doing, by the way). Our test on matrices I believe is on Friday, or Wednesday. I can't remember if it's been settled ( sorry :( ).

Anyhow, the problem is as follows:

As we all know by now, our beloved Manitoba Moose are a team that is prong to streaky play. In other words, when they are winning, it seems like they will never lose again. On the other hand, there are times when they are losing and it seems like they will never win another game! Being the statistical sport that hockey is, Joe scoreboard (very lame :P) has made the following observations concerning the Moose. That is, when the Moose win a game, there is a 70% chance that they will win their next game. However, when the moose lose a game, they have only a 40% chance of winning their next game. Use this data to answer the following questions.

1. Write a transition matrix describing the situation above.

W L
T = W [.70 .30] The sum of each row must equal 1 (100%).
L [.40 .60]

2) Suppose that the moose won their first game of the season. What would the initial distribution (initial state) matrix?

D0[1 0] 1 represents that they won, while 0 represents that there is no chance they lost, because well, they won.

3) Given the conditions from 1 and 2, find the Moose's probable winning percentage after 1,2, 3, 4, 5, and 10 games. Mulitplythe matrix in question 2 by the one in number 1.

For example, Do10 = D0T10 ... What does D10 represent? ... For this one, I believe it means the number of games.

1) [.7 .3]
2) [.61 .39]
3) [.583 .417]
4) [.5759 .4251]
5) [.5725 .4275]
10) [.5714] [.4286]

Having played 10 games, could the Moose actually have a record represented by this posibility?
The answer is that they can't, because after 10 games you can't have won 57% of them.

4) Now, recalculate example 3 with the condition that the Moose lost their first game. Complete the chart below and compare the results. *To get the probabilities from having lost the first game, just change the state matrix from [1 0] to [0 1]

Winning percentage having won or lost first game.
W L
1 game .70 .4
2 games .61 .52
3 games .583 .556
4 games .5749 .5568
5 games .5725 .5700
10 games .5714 .5714

5) Notice that there is very little difference between the results after 5 games and after 10 games. What do these percentages represent?
These percentages represent the matrix stabilizing, meaning they can expect to win 57% of all the games they play.

7) The Moose play an 80 game regular season. Predict their final won-loss record assuming that they win their first game and also if they lose their first game. How do the records compare? In this example, how important is the initial state?

The records are the same, [.5714 .4286] .... This means that regardless of how their season begins, they may win 57% of all games they play. Therefore, the initial state doesn't really mean anything over time.


I am very sorry that it took me so long to post this (I also apologize for the length!). I hope I explained everything clearly and properly. Monday's scribe will be Mohamed.




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Cait, no appologies necessary. You've done an outstanding job!!

The amount of detail you have included is really going to help your classmates study for the test and, later, for the exam. This is really well done.

You have raised the art of the scribe to a new level. Bravo!

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