Question

According to the 2017 SAT Suite of Assessments Annual Report, the average ERW (English, Reading, Writing) SAT score in Florida was 520. Assume that the scores are Normally distributed with a standard deviation of 100. Answer the following including an appropriately labeled and shaded Normal curve for each question.

a) What is the probability that an ERW SAT taker in Florida scored 500 or less?

b) What percentage of ERW SAT takers in Florida scored between 500 and 650?

c) What ERW SAT score would correspond with the 40th percentile in Florida?

Answer 1

a)

X ~ N ( µ = 520 , σ = 100 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 500 ) = P ( Z < 500 - 520 ) / 100 )
= P ( Z < -0.2 )
P ( X < 500 ) = 0.4207 (From Z table)

b)

Given :- = 520 , = 100 )
We convet this to Standard Normal as
P(X < x) = P( Z < ( X - ) / )
P ( 500 < X < 650 ) = P ( Z < ( 650 - 520 ) / 100 ) - P ( Z < ( 500 - 520 ) / 100 )
= P ( Z < 1.3) - P ( Z < -0.2 )
= 0.9032 - 0.4207 (From Z table)
= 0.4825
= 48.25%

c)

We have to calculate P(X < x) = ?.
X ~ N ( µ = 520 , σ = 100 )
P ( X < x ) = 40% = 0.4
To find the value of x
Looking for the probability 0.4 in standard normal table to calculate critical value Z = -0.2533
Z = ( X - µ ) / σ
-0.2533 = ( X - 520 ) / 100
X = 494.67