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 1117 and a standard deviation of 194. Scores on the ACT test are normally distributed with a mean of 21.9 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 77-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 1544, find the equivalent ACT score.
ACT score =
Round answer to 1 decimal place.

Answer:

The actual SAT score is 1260.6.

The equivalent ACT score for this student(77th percentile) is 24.7.

The equivalent ACT score to a SAT score of 1544 is 30.26.

Step-by-step explanation:

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

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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:

[tex]\mu = 1117, \sigma = 194[/tex]

ACT:

[tex]\mu = 21.9, \sigma = 3.8[/tex]

If a student gets an SAT score that is the 77-percentile, find the actual SAT score.

This is X when Z has a pvalue of 0.77. So X when Z = 0.74.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]0.74 = \frac{X - 1117}{194}[/tex]

[tex]X - 1117 = 0.74*194[/tex]

[tex]X = 1260.6[/tex]

The actual SAT score is 1260.6.

What would be the equivalent ACT score for this student?

X when Z = 0.74, using [tex]\mu = 21.9, \sigma = 3.8[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]0.74 = \frac{X - 21.9}{3.8}[/tex]

[tex]X - 21.9 = 0.74*3.8[/tex]

[tex]X = 24.7[/tex]

The equivalent ACT score for this student(77th percentile) is 24.7.

If a student gets an SAT score of 1544, find the equivalent ACT score.

First we find the z-score of the SAT, then we find the equivalent ACT score.

Z-score of SAT:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{1544 - 1117}{194}[/tex]

[tex]Z = 2.2[/tex]

Equivalent ACT:

X when Z = 2.2. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]2.2 = \frac{X - 21.9}{3.8}[/tex]

[tex]X - 21.9 = 2.2*3.8[/tex]

[tex]X = 30.26[/tex]

The equivalent ACT score to a SAT score of 1544 is 30.26.

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