Question 84753: The scores for a standardized reading test are found to be normally distributed with a mean of 500 and a standard diviation of 60. If the test is given to 900 students, how many are expected to have scores between 500 and 620?
A)306
b) 432
c)459
d)612
Answer by Scriptor(36)   (Show Source):
You can put this solution on YOUR website! Hello,
Let's calculate the probability that one student has a score between 500 and 600. This is P(500< X < 620)
I use the notation X for the score:
P(500 < X < 620)
= P( 500 - 500 < X - 500 < 620 - 500) (subtract 500 on all sides)
= P(0 < X - 500 < 120)
= P (0/60 < (X-500)/60 < 120/60) (divide bij 60 on all sides)
= P(0 < (X-500)/60 < 2)
Since X is normally distributed with mean 500 and std 60, we have that
(X-500)/60 is standardnormally distibuted:
X~N(500,60²) <=> (X-500)/60 ~ N(0,1)
This means that:
= P(0 < (X-500)/60 < 2) = Phi(2) - Phi(0)
The values of Phi(2) and Phi(0) can be looked up, they are:
Phi(2)= 0.9772 and Phi(0)= 0.5
=> P(0 < (X-500)/60 < 2) = 0.9972 - 0.5
= 0.4772
~ 0.48
This means that there is 48% that the score will be wetween 500 and 620.
For 900 students we thus expect that we expect 0.48*900=432 students with a score between 500-620.
Answer: 432
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