Question

The College Board National Office recently reported that in 2011–2012, the 547,038 high school juniors who took the ACT achieved a mean score of 515 with a standard deviation of 129 on the mathematics portion of the test (http://media.collegeboard.com/digitalServices/pdf/research/2013/TotalGroup-2013.pdf). Assume these test scores are normally distributed.

1. What is the probability that a high school junior who takes the test will score at least 590 on the mathematics portion of the test? If required, round your answer to four decimal places.
   
   P (x ≥ 590) =
   
2. What is the probability that a high school junior who takes the test will score no higher than 510 on the mathematics portion of the test? If required, round your answer to four decimal places.
   
   P (x ≤ 510) =
   
3. What is the probability that a high school junior who takes the test will score between 510 and 590 on the mathematics portion of the test? If required, round your answer to four decimal places.
   
   P (510 ≤ x ≤ 590) =
   
4. How high does a student have to score to be in the top 10% of high school juniors on the mathematics portion of the test? If required, round your answer to the nearest whole number.
   

Answer 1

Given that the ACT achieved a mean, score of 515 with a standard deviation,  of 129 on the mathematics portion of the test.

and the distribution is normal hence using Z score the probability will be calculated as:

a) P( X>=590)

Z at X=590

Probability is calculated using the Z table shown below as

P(X>=590)=P(Z>0.58)

=0.2810

b) P(X<=510) now Z at X=510

so, P(X<=510)=P(Z<-0.04)

=0.4880.

c) P(510<X<590)

Since Zat 590 is 0.58 and at X=510 Z=-0.04 hence probability will be

P(-0.04<Z<0.58)

=0.7190-0.4880

=0.231

d) The boundary for the top 10 will be calculated from the Z table shown below as

Z=1.29

Hence using Z formula