Calculate the percentile correspond and the percentile of the sampling distribution The SAT placement test for admission to American universities had an average of μ = 1026 and a standard deviation σ = 209. The distribution of marks is approximately normal. a) Ramon had a grade of 1100. If the notes have a normal distribution, to what percentile does Ramon's note correspond? b) Now consider the average X¯ of the scores of 70 randomly selected students. If X¯ = 1100, at what percentile of the sampling distribution of X¯ does this result correspond? c) Which of your two calculations, a) or b), is less precise since SAT notes do not follow exactly a normal law? Here is my approach: a) I calculated z-score = 0.354 then I found that the proportion correspond is 0.6368. So what is the percentile correspond? 63.68 or 64? b) I obtained z-score =2.96 so the proportion is 0.9985. Then what is the percentile?

(a) Computation using R statistical software.


pnorm(1100, 1026, 209)
[1] 0.6383557


In this testing context, percentile is defined as 'percent below', so I'd say 63.83%, often rounded to 64% (except that rounding to integers is unusual for results above 90%).

(b) Computation in R: I'd say 99.85%.


pnorm(1100, 1026, 209/sqrt(70))
[1] 0.9984734


Note: If you are getting probabilities from printed tables of the normal CDF, then you may need to round in order to use the tables. Thus it is possible for you to get slightly different results than the exact ones I got from R.

(c) I suppose you are expected to say that (b) is more accurate because the Central Limit Theorem will tend to make the average of 70 scores more nearly normal than an individual score. (A quibble in this practical application is that fitting a normal model is more difficult in the tails than near the center of the distribution.)