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 999 and a standard deviation of 199. Scores on the ACT test are normally distributed with a mean of 21.6 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 62-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. B. If a student gets an SAT score of 1563, find the equivalent ACT score. ACT score = Round answer to 1 decimal place.

Answer 1

A.

SAT score = 1060

ACT score = 23.2

B.

ACT score = 36.3

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

SAT:

`\mu = 999, \sigma = 199`

If a student gets an SAT score that is the 62-percentile, find the actual SAT score.

This is X when Z has a pvalue of 0.62. So X when Z = 0.305.

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

`0.305 = (X - 999)/(199)`

`X - 999 = 0.305*199`

`X = 1059.7`

Rounding to the nearest whole number.

SAT score = 1060

ACT:

`\mu = 21.6, \sigma = 5.2`

The equivalent score is X when Z = 0.305.

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

`0.305 = (X - 21.6)/(5.2)`

`X - 21.6 = 0.305*5.2`

`X = 23.19`

So

ACT score = 23.2

B. If a student gets an SAT score of 1563, find the equivalent ACT score

Z-score for the SAT score.

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

`Z = (1563 - 999)/(199)`

`Z = 2.83`

Equivalent ACT:

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

`2.83 = (X - 21.6)/(5.2)`

`X - 21.6 = 2.83*5.2`

`X = 36.3`

ACT score = 36.3