Problem:A law school has the following data about their students. AVE LSAT scores are about 650 pts with an SD of about 80 pts. The AVE of the first year test scores is about 65 points with an SD of about 10 pts. The correlation coefficient is about 0.6. |
Question1:
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Assuming that the scatter plot is football-shaped what is the
average first year score for those students scoring 730 on the
LSAT test?
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Question2:
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What percentage of the students with 730 pts in the LSAT will
have first year scores in the interval 61 to 81?
Here are the answers.
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Answer1:
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RECALL to predict y from x do:
1) transform x to standard units. 2) Multiply (x in SU) by the correlation coefficient r. 3) The answer is the result of 2) in SU for y. So let's do 1, 2 and 3. We have, |
> aveLSAT := 650: sdLSAT := 80: aveFY := 65: sdFY := 10: r := 0.6:
> xLSAT := 730:
| so xLSAT in standard (LSAT pts) units is, |
> xSU := (xLSAT - aveLSAT)/sdLSAT;
xSU := 1
| and the answer in SU for first year scores is: |
> ySU := xSU*r;
ySU := .6
| transforming back to original first year scores pts... |
> yFY := aveFY + ySU*sdFY;
yFY := 71.0
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Let's do the next.
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Answer2:
| Those students that scored 730 in the LSAT will have first year scores that average out to 71 pts. with an SD about the r.m.s. error for the regression of FY on LSAT. So lets first compute the rms. |
> rmsFY := 'sqrt(1-r^2)'*sdFY;
2
rmsFY := 10 sqrt(1 - r )
| which evaluates to, |
> evalf(rmsFY,2);
8.0
| Thus, the percentage of the studens (who score 730 in the LSAT) who have FY scores in the interval (61,81) will be approx. the area under the normal curve between the limits in su. We have |
> LowSU := (61 - 71)/rmsFY; upSU := (81 - 71)/rmsFY;
LowSU := -1.250000000
upSU := 1.250000000
| Looking up the table with z=1.25 we get an area of about 79% and this is the answer. |