normal distribution, with µ = 3.2 and σ = 0.75. (a) What is the maximum score that can be in the bottom 10% of scores? (b) Between what two values does the middle 76% of the scores lie?

Question

The test scores for the quantitative reasoning section of the Graduate Record Examination (GRE) can be approximated by a normal distribution, with µ = 3.2 and σ = 0.75. For the problems below, round your final answers to two decimal places.  (a) What is the maximum score that can be in the bottom 10% of scores?  (b) Between what two values does the middle 76% of the scores lie?

Answer 1

(a) Z=(X-µ)/σ, so we need the value of Z corresponding to the lower 10%. -1.282 gives us about 0.1.

-1.282=(X-3.2)/0.75, so X=3.2-0.75(1.282)=2.24 approx.

(b) The middle 76% leaves 24% to be shared equally between the tails of the distribution. So we need the Z value for 2-tail 12%. Z=-1.175 gives the lower 12% so Z=1.175 gives 88% (top 12%). From these two values we can work out the X values. X₁=3.2-0.75(1.175)=2.32, X₂=3.2+0.75(1.175)=4.08. These two values enclose 76% of the scores.