Question

According to the College Board, SAT critical reading scores from the 2014 school year for high school students in the United States were normally distributed with a mean of 497 and a standard deviation of 115. Use a standard normal table such as this one to determine the probability that a randomly chosen high school student who took the SAT in 2014 will have a critical reading SAT score between 450 and 750 points Give your answer as a percentage rounded to one decimal placeP=_____%

Answer 1

`P=` 64.5 %

Explanation:

Let's start defining the random variable X.

X : '' SAT critical reading scores from the 2014 school year for high school students in the United States ''

We know that X ~ N (μ,σ)

Where μ is the mean and σ is the standard deviation.

⇒ X ~ N (497,115)

If we want to calculate probabilities related to X we need to standardized the random variable. We do this by subtracting the mean to X and then dividing by the standard deviation. This new random variable will be ''Z'' and Z ~ N (0,1)

We can find the probabilies of ''Z'' in any standard normal table.

The cumulative distribution of ''Z'' is the function Φ where :

`P(Z\leq a)=` Φ(a)

Now, we need to calculate the following probability :

`P(450<X<750)`

If we standardized this :

`P((450-497)/(115)<(X-497)/(115)<(750-497)/(115))`

We know that
`(X-497)/(115)` ≅ Z ⇒

`P((450-497)/(115)<Z<(750-497)/(115))=P(-0.41<Z<2.2)` ⇒

`P(-0.41<Z<2.2)=` Φ(2.2) - Φ(-0.41) =
`0.9861-0.3409=0.6452`

⇒ 64.52% ≅ 64.5%

We find that the probability (given as a percentage) is 64.5%