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 1070 and a standard deviation of 204. Scores on the ACT test are normally distributed with a mean of 19.1 and a standard deviation of 5.2. It is assumed that the two tests measure the same aptitude, but use different scales.

(A) If a student gets an SAT score that is the 51-percentile, find the actual SAT score. Round answer to a whole number. SAT score =
(B) What would be the equivalent ACT score for this student? Round answer to 1 decimal place. ACT score =
(C) If a student gets an SAT score of 1417, find the equivalent ACT score. Round answer to 1 decimal place. ACT score =

Answer 1

a) SAT score = 1075

b) ACT score = 19.2.

c) ACT score = 27.9.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value 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. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

(A) If a student gets an SAT score that is the 51-percentile, find the actual SAT score

SAT scores have mean 1070 and standard deviation 204, so
`\mu = 1070, \sigma = 204`

51th percentile means that Z has a p-value of 0.51, so Z = 0.025. The score is X. So

`Z = (X - \mu)/(\sigma)`

`0.025 = (X - 1070)/(204)`

`X - 1070 = 0.025*204`

`X = 1075`

SAT score = 1075.

(B) What would be the equivalent ACT score for this student?

ACT scores have mean of 19.1 and standard deviation of 5.2, which means that
`\mu = 19.1, \sigma = 5.2`. The equivalent score is X when Z = 0.025. So

`Z = (X - \mu)/(\sigma)`

`0.025 = (X - 19.1)/(5.2)`

`X - 19.1 = 0.025*5.2`

`X = 19.2`

ACT score = 19.2.

(C) If a student gets an SAT score of 1417, find the equivalent ACT score.

Z-score for the SAT:

`Z = (X - \mu)/(\sigma)`

`Z = (1417 - 1070)/(204)`

`Z = 1.7`

Equivalent score on the ACT:

`Z = (X - \mu)/(\sigma)`

`1.7 = (X - 19.1)/(5.2)`

`X - 19.1 = 1.7*5.2`

`X = 27.9`

ACT score = 27.9.