Matrix Algebra

A study of students taking a 20-question exam ranked their progress from one testing peroid to the next. Students scoring 0 to 5 form group 1, those scoring 6 to 10 form group 2, those scoring 11 to 15 form group 3, and those scoring 16 to 20 form gropu 4. The transition matrix below shows the result.

0.147 0.127 0.31 0.416

0.339 0.12 0.21 0.331

0.339 0.355 0.047 0.259

0 0.173 0.498 0.329

Suppose all students are initially in group 1. When a student reaches group 4, the student is said to have masered the material and is no longer tested, so the student stays in that group forever. Find the number of testing periods you would expect for at least 70% of the students to have mastered the material

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The last row of the transition matrix is not correct. Once you are in group 4, you stay in group 4 forever. Thus the row of the transition matrix must be:

0 0 0 1

That is:

the probability of going from group 4 to group 1 is 0

the probability of going from group 4 to group 2 is 0

the probability of going from group 4 to group 3 is 0

the probability of going from group 4 to group 4 is 1

(This is what the row should be, to match the scenario you describe)

Also, the third row of your matrix seems suspect. (Not necessarily wrong, but it seems unusual given your scenario):

the probability of going from group 3 to group 1 is .339

the probability of going from group 3 to group 2 is .355

the probability of going from group 3 to group 3 is .047

the probability of going from group 3 to group 4 is .259

Presumably the students are learning things, so, for the most part their scores should be rising. Here, almost 70% of the students in group 3 score lower on their next test. And 26% score higher, but only 5% score about the same. This seems a little unusual for this scenario.

My guess is you've got the transition matrix for a different question.

So, I will use the following matrix for this explanation:

`[[.147, .127, .31,.416],[.339,.12,.21,.331],[.339,.355,.047,.259],[0,0,0,1]]`

I've kept your third row, but I've replaced the fourth row with one that represents an "end state" (that is, once you reach group 4, you stay there).

Call this matrix A.

Note that .416 represents the probabilty that you start in group 1, and end in group 4, after 1 test. (The first row represents starting in group 1, the 4th column represents ending up in group 4).

A*A=

`[[.17,.113,.087,.599],[.162,.111,.14,.566],[.152,.086,.161,.497],[0,0,0,1]]`

This matrix represents the transition probabilities after two tests. That is, the probability of moving from group 1 to group 4 after 2 tests is .599

`A^3` =

`[[.093,.057,.08,.73],[.109,.07,.08,.706],[.106,.071,.073,.63],[0,0,0,1]]`

Thus, the probabilty of a student going from group 1 to group 4 after 3 tests is .73.

So we would expect about 73% of the students to make it to group 4 after 3 tests.

So 3 tests is the answer.

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