Suppose scores on the mathematics section of the SAT follow a normal distribution with mean 540 and standard deviation 120. ACT math scores are normally distributed with a mean of 18 and a standard deviation of 8. What score on the ACT is equivalent to a score of 750 on the SAT
To find the equivalent score on the ACT for a score of 750 on the SAT, we can use z-scores.
A z-score measures the number of standard deviations an individual value is from the mean of a distribution. It is calculated using the formula:
z = (x - μ) / σ
Where:
- x is the individual value (score on the SAT)
- μ is the mean of the distribution (mean SAT score)
- σ is the standard deviation of the distribution (standard deviation of SAT scores)
First, let's calculate the z-score for the score of 750 on the SAT:
z_sat = (750 - 540) / 120
= 210 / 120
= 1.75
Now, we need to find the equivalent score on the ACT using the z-score and the parameters for the ACT math scores:
Equivalent ACT score = (z_act * σ_act) + μ_act
Where:
- z_act is the z-score we calculated for the SAT score on the ACT distribution
- σ_act is the standard deviation of the ACT math scores
- μ_act is the mean of the ACT math scores
Substituting the given values:
Equivalent ACT score = (1.75 * 8) + 18
= 14 + 18
= 32
So, the equivalent ACT score for a score of 750 on the SAT is 32.