Math and verbal SAT scores are each distributed normally with N (500,10000). (a)
ID: 3302699 • Letter: M
Question
Math and verbal SAT scores are each distributed normally with N (500,10000).
(a) What fraction of students scores above 750? Above 600? Between 420 and 530? Below 480? Above 530?
(b) If the math and verbal scores were independently distributed, which is not the case, then what would be the distribution of the overall SAT score? Find its mean and variance.
(c) Next, assume that the correlation coefficient between the math and verbal scores is 0.75. Find the mean and variance of the resulting distribution.
(d) Finally, assume that you had chosen 25 students at random who had taken the SAT exam. Derive the distribution for their average math SAT score. What is the probability that this average is above 530? Why is this so much smaller than your answer in (a)?
Can someone show the work for how you got your answers?
Explanation / Answer
here men =500
and std deviation =(10000)1/2 =100
therefore a) fraction of students scores above 750 =P(X>750)=P(Z>(750-500)/100)=P(Z>2.5)=0.0062
Above 600 =P(X>600) =P(Z>1) =0.1587
Between 420 and 530 =P(420<X<530)=P(-0.8<Z<0.30)= 0.6179-0.2119 =0.4061
Below 480 =P(X<480)=P(Z<-0.20)=0.4207
Above 530 =P(X>530)=P(Z>0.3)=0.3821
b) distribution of the overall SAT score would be normal with mean =(500+500)/2 =500
and variance =(1002+1002)/4 =5000
c) mean =500
for Var(aX+bY) =a2Var(X)+b2Var(Y)+2ab*(Var(X)*Var(Y))1/2 *correlation coefficient
variance =1002/4+ 1002/4+2*(1/2)(100*100)*correlation coefficent =12500
d) here std error of mean =std deviation/(n)1/2 =100/(25)1/2 =20
probability that this average is above 530 =P(X>530)=P(Z>(530-500)/20) =P(Z>1.5)=0.0668