SAT scores: The College Board reports that in a recent year, the mean mathematics SAT score was 514, and the standard deviation was 118. A sample of 65 scores is chosen.

Question

SAT scores: The College Board reports that in a recent year, the mean mathematics SAT score was 514, and the standard deviation was 118. A sample of 65 scores is chosen.

a.	What is the probability that the sample mean score is less than 500?

Answer 1

When you are taking a sample of greater than 30, you can assume that the distribution of sample means is Normal - this is known as the Central Limit Theorem. The mean and standard deviation is found mathematically when the sample is chosen at random from the population.

That being said, you are given the Mean µ=514 and the Standard Deviation σ=118 of SAT Scores (each is a value of x from all scores X (the population). You are taking a sample of size n=65 and looking at the average of the sample X-bar.

What is the probability that the sample mean score is less than 500 = P(X-bar < 500)

The distribution of sample means, X-bar, is distributed Normally when n > 30 with a mean of µ=514 (sane as the population, which makes sense) and standard deviation related to σ and n which is σ/√n = 118/√65 = 14.636 (also makes sense to have n on the bottom since as n gets larger, the spread of the averages gets closer to the mean).

Since we know the shape, mean and standard deviation of the distribution, we can convert to the Standard Normal Distribution Z and find the probability.

P(X-bar < 500) = P((X-bar-514)/14.636 < (500-514)/14.636) = P(Z < -0.96) = Φ(-0.96) = 0.1685