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So I know this may be basic but I'm having trouble piecing together the right formulas and logic to understand this. Most of the Z-tables online seem to have different information than the one provided to us so I've left the table below. Its a centered Z-table and not a left tail one.

For first year students at a university the correlation between SAT scores and first year GPA is 0.60. The scatter diagram is football shaped. Predict the first year GPA for a student whose percentile rank on the SAT was 30%.

In order to get the proper z-score I subtracted the remaining percent twice from 100%. I dont follow the logic behind this but apparently it works. |100 - 70 - 70| = 40%. The z score corresponding to 40% of the area on our table is ~.53

I then multiplied by the correlation to get ~0.32 (this makes sense to me). This corresponds to about 25%. I'm not quite sure how to then get this number back into percentile. The answer is 38%.

Thanks for the help! Z-Table

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  • $\begingroup$ Note on the table: The central areas have been multiplied by 100 to give percentages, okay, I can get that, but for a standard normal the "heights" are nonsense since they're not percentages of anything. They should not be multiplied by 100. $\endgroup$ – Glen_b Sep 28 '17 at 3:43
  • $\begingroup$ The correct calculation for your |100-70-70| is 100-30-30 = 40 (no absolute value involved, no subtracting 30 from 100). from 100% you subtract the portion in the left tail (30%) and the same proportion in the right tail (another 30%) leaving the part in the middle (40%). The correct calculation is more direct, easier to understand and faster, but will yield the same result as your approach that doesn't have the same simple justification. $\endgroup$ – Glen_b Sep 28 '17 at 3:46
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You almost got there.

Your 0.32 would more accurately be 0.315 (for number of sd's below the mean). This gives a central area of about 25% and a left tail area of about 38%. More accurately, (100-24.8)/2=37.6 (Your 24.8 is 100-?-? = 24.8 and you want to figure out what that ? value would be)

Plot of bivariate distribution illustrating the steps in the question and the last step here (1-0.248)/2 = 0.376

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