Question

(1 point) The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean μ=534 and standard deviation σ=27.3. (a) What is the probability that a single student randomly chosen from all those taking the test scores 539 or higher?

Answer 1

Probability that a single student randomly chosen from all those taking the test scores 539 or higher is 0.5714.

Answer 2

Probability that a single student randomly chosen from all those taking the test scores 539 or higher is 0.5714.

Explanation:

We are given that the scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean μ = 534 and standard deviation σ = 27.3.

Let X = scores of students on the SAT college entrance examinations

SO, X ~ N(
`\mu = 21.2,\sigma^(2) = 6.2^(2)`)

The z-score probability distribution is given by ;

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

where,
`\mu` = mean score = 534

`\sigma` = standard deviation = 27.3

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, the probability that a single student randomly chosen from all those taking the test scores 539 or higher is given by = P(X
`\geq` 539)

P(X
`\geq` 539) = P(
`(X-\mu)/(\sigma)`
`\geq`
`(539-534)/(27.3)` ) = P(Z
`\geq` 0.18)

= 0.5714 {using z table}

Therefore, probability that a single student scores 539 or higher is 0.5714.