A study was conducted on students from a particular high school over the last 8 years....

Question

A study was conducted on students from a particular high school over the last 8 years. The following information was found regarding standardized tests used for college admitance. Scores on the SAT test are normally distributed with a mean of 995 and a standard deviation of 195. Scores on the ACT test are normally distributed with a mean of 22.1 and a standard deviation of 3.8. It is assumed that the two tests measure the same aptitude, but use different scales.

If a student gets an SAT score that is the 24-percentile, find the actual SAT score.

SAT score = 

Round answer to a whole number.

What would be the equivalent ACT score for this student?

ACT score = 

Round answer to 1 decimal place.

If a student gets an SAT score of 1288, find the equivalent ACT score.

ACT score = 

Round answer to 1 decimal place.

Answer 1

The key to this problem is standardizing the normal distribution. For notation, let N(mu, s^2) represent a normal distribution with mean mu and standard deviation s (variance is s^2).

For the SAT scores, you are given the distribution N(995, 195^2). For the ACT scores, you are given the distribution N(22.1, 3.8^2).

To standardize the normal distribution, you will need to calculate the z-score: (X - mu)/s. X is either the SAT or ACT score, depending on the problem. The z-score follows the distribution N(0, 1^2), of which you can use a z-table (or statistical software) to calculate the cumulative distribution function (CDF).

For the SAT score in the 24th percentile, you want to find the z-score such that the CDF evaluates to 0.24. On a z-table, this value should be somewhere between -0.70 and -0.71 (try to figure out how to get this value). Once you have the z-score, since you know mu = 995 and s = 195, you can solve for the SAT score.

For the equivalent ACT score, the procedure is exactly the same, but with mu = 22.1 and s = 3.8. By standardizing, it is easier to compare normal distributions.

For the SAT score of 1288 and the equivalent ACT score, your goal is to find the corresponding percentile. You can first calculate the z-score. You can then use the same z-score and solve for X with the ACT mean and standard deviation.