Statistics question using a normal distribution

Question

The SAT and ACT college entrance exams are taken by thousands of students each year. The scores on the exam for any one year produce a histogram that looks very much like a normal curve. Thus, we can say that the scores are approximately normally distributed. In recent years, the SAT mathematics scores have averaged around 480 with standard deviation of 100. The ACT mathematics scores have averaged around 18 with a standard deviation of 6.

 

 

a. An engineering school sets 550 as the minimum SAT math score for new students. What percent of students would  score less than 550 in a typical year?

 

b. What would the engineering school set as comparable standard on the ACT math test?

 

c. What is the probability that a randomly selected student will score over 700 on the SAT math test?

Answer 1

a.

 

Standard score: z = (x - μ)/σ = (550 - 480) / 100 = 0.7

 

Now standard normal table to find Φ(0.7) = 0.7580

 

So about 75.8% will score below 550 on SAT math.

 

 

b.

 

We still want z = 0.7 so

 

(x - 18) / 6 = 0.7

 

x - 18 = 4.2

 

x = 22.2

 

This cutoff means that they should accept only students scoring at least 23 on ACT math.

 

 

c.

 

Standard score: z = (x - μ)/σ = (700 - 480) / 100 = 2.20

 

Using the table again:

 

P(x > 700) = 1 - P(x ≤ 700) = 1 - Φ(2.2) = 1 - 0.9861 = 0.0139

 

So There is a 1.39% chance that a student will score more than 700 on SAT math.