Changing the mean and standard deviation of a Normal distribution by a moderate amount can greatly change the percent of observation in the tails.

Question

Changing the mean and standard deviation of a Normal distribution by a moderate amount can greatly change the percent of observation in the tails. Suppose that a college is looking for applicants with SAT math scores 700 and above.

(a) In 2007, the scores of students in Colorado on the math SAT followed the N(533, 116) distribution. What percent of these students scored 700 or better?

(b) The scores of students in Delaware that year had the N(499, 110) distribution. What percent of these students scored 700 or better

Answer 1

To find the percentage of students scoring 700 or better on the SAT math section for each state, we'll use the standard normal distribution and convert the scores to z-scores.

The formula to convert a raw score (𝑋X) to a z-score (𝑍Z) is:

𝑍=𝑋−𝜇𝜎Z=σX−μ​

Where:

1. 𝑋X is the raw score
2. 𝜇μ is the mean of the distribution
3. 𝜎σ is the standard deviation of the distribution

Then, we'll use a standard normal distribution table or a calculator to find the percentage of students with z-scores greater than or equal to the z-score corresponding to a raw score of 700.

For Colorado: 𝑍=700−533116Z=116700−533​ 𝑍=167116≈1.44Z=116167​≈1.44

For Delaware: 𝑍=700−499110Z=110700−499​ 𝑍=201110≈1.82Z=110201​≈1.82

Now, using a standard normal distribution table or a calculator, we find the percentage of students with z-scores greater than or equal to 1.44 for Colorado and 1.82 for Delaware.

(a) For Colorado: The percentage of students scoring 700 or better is approximately 7.56%.

(b) For Delaware: The percentage of students scoring 700 or better is approximately 3.70%.