Question

For a certain standardized placement test, it was found that the scores were normally distributed, with a mean of 250 and a standard deviation of 30. Suppose that this test is given to 1000 students. (Recall that 34% of z-scores lie between 0 and 1, 13.5% lie between 1 and 2, and 2.5% are greater than 2.)

(a) How many are expected to make scores between 220 and 280?
students

(b) How many are expected to score above 310?
students

(c) What is the expected range of all the scores?

Answer 1

here we use standard normal z=(x-mean)/sd

(a) answer is 683

for x=220, z=(220-250)/30=-1

for x=280, z=(280-250)/30=1

P(220<X<280)=P(-1<Z<1)=P(Z<1)-P(Z<-1)=0.8413-0.1587=0.6826

so expected number to make scores between 220 and 280 is 1000*0.6826=682.6 ( next whole number is 683)

(b)answer is 23

for x=310, z=(310-250)/30=2

P(X>310)=P(Z>2)=1-P(Z<=2)=1-0.9773=0.0227

expected number of student to score above 310 is 0.0227*1000=22.7 ( next whole number is 23)

(c) answer is 160 to 340

almost all the students will come within 3 standard deviation

so require range will be( mean-3*sd, mean+3sd)=(250-30*3, 250+30*3)=(160, 340)