Eleanor scores 680 on the mathematics part of the SAT. The distribution of SAT math scores in recent years has been Normal with mean 513 and standard deviation 81. Gerald takes the ACT Assessment mathematics test and scores 27. ACT math scores are Normally distributed with mean 22.7 and standard deviation 4.What is Elanor's standardized score?

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Part a

Let X the random variable that represent the SAT scores of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(513,81)[/tex]

Where [tex]\mu=513[/tex] and [tex]\sigma=81[/tex]

For this case we can calculate the z score with the following formula:

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

And if we replace we got:

[tex] z =\frac{680-513}{81}=2.06[/tex]

So the Eleanor score is 2.06 deviations above the mean.

Part b

Let X the random variable that represent the ACT scores of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(22.7,4)[/tex]

Where [tex]\mu=22.7[/tex] and [tex]\sigma=4[/tex]

For this case we can calculate the z score with the following formula:

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

And if we replace we got:

[tex] z =\frac{27-22.7}{4}=1.08[/tex]

So the Gerald score is 1.08 deviations above the mean.