Question

(a) In 2000, the scores of students taking SATs were normally distributed with mean 1019 and standard deviation 209. (i) What percent of all students had the SAT scores of at least 820? [3 marks] (ii) What percent of all students had the SAT scores between 720 and 820? [3 marks] (iii) How high must a student score in order to place in the top 20% of all students taking the SAT? [4 marks]

Answer 1

$\mu =1019,\sigma =209$

i) $Z=\frac{X-\mu }{\sigma }=\frac{820-1019}{209}=\frac{-199}{209}=-0.95$

$P(X\geq 820)=P(Z\geq -0.95)=0.8295$

ii)$Z_{1}=\frac{X_{1}-\mu }{\sigma }=\frac{720-1019}{209}=\frac{-299}{209}=-1.43$

$Z_{2}=\frac{X_{2}-\mu }{\sigma }=\frac{820-1019}{209}=\frac{-199}{209}=-0.95$

iii) Z-score of top 20% is $Z=0.84$

Let required score be X

$X=\mu +Z\sigma =1019+0.84\times 209=1195$