Question

There are two major tests of readiness for college: the act and the sat. act scores are reported on a scale from 1 to 36. the distribution of act scores in recent years has been roughly normal with mean μ = 20.9 and standard deviation σ = 4.8. sat scores are reported on a scale from 400 to 1600. sat scores have been roughly normal with mean μ = 1026 and standard deviation σ = 209. joe scores 1351 on the sat. assuming that both tests measure the same thing, what score (±1) on the act is equivalent to joe's sat score? .

Answer 1

Solution: We are given:

ACT scores follow normal distribution with $\mu=20.9,\sigma =4.8$

SAT scores follow normal distribution with $\mu=1026,\sigma=209$

Now, let's find the z score corresponding to Joe's SAT score 1351.

$z=\frac{x-\mu}{\sigma}$

        $=\frac{1351-1026}{209}$

        $=\frac{325}{209}$

        $=1.56$

Therefore, Joe's SAT score is 1.56 standard deviations above the mean.

Now, we have find the Joe's ACT score, which will be 1.56 standard deviations above the mean.

Therefore, we have:

$z=\frac{x-\mu}{\sigma}$

$1.56=\frac{x-20.9}{4.8}$

$1.56 \times 4.8 = x - 20.9$

$7.488=x-20.9$

$x=20.9+7.488$

$x=28.388 \approx 28.4$

Therefore, Joe's equivalent ACT score to SAT score 1351 is 28.4