Question

Math scores on the SAT exam are normally distributed with a mean of 514 and a standard deviation of 118. If a recent test-taker is selected at random, what is the probability the student scored between 455 and 573 on the exam?

Answer 1

Probability that the student scored between 455 and 573 on the exam is 0.38292.

Explanation:

We are given that Math scores on the SAT exam are normally distributed with a mean of 514 and a standard deviation of 118.

Let X = Math scores on the SAT exam

So, X ~ Normal(
`\mu=514,\sigma^(2) =118^(2)`)

The z score probability distribution for normal distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = population mean score = 514

`\sigma` = standard deviation = 118

Now, the probability that the student scored between 455 and 573 on the exam is given by = P(455 < X < 573)

P(455 < X < 573) = P(X < 573) - P(X
`\leq` 455)

P(X < 573) = P(
`(X-\mu)/(\sigma)` <
`(573-514)/(118)` ) = P(Z < 0.50) = 0.69146

P(X
`\leq` 2.9) = P(
`(X-\mu)/(\sigma)`
`\leq`
`(455-514)/(118)` ) = P(Z
`\leq` -0.50) = 1 - P(Z < 0.50)

= 1 - 0.69146 = 0.30854

The above probability is calculated by looking at the value of x = 0.50 in the z table which has an area of 0.69146.

Therefore, P(455 < X < 573) = 0.69146 - 0.30854 = 0.38292

Hence, probability that the student scored between 455 and 573 on the exam is 0.38292.