Question

2. Many students in the Midwest take both the SAT I and the ACT. One year, the distribution of SATI

math scores was approximately normal with mean 510 and standard deviation 110. The distribution
of ACT math scores was approximately normal with mean 20.1 and standard deviation 4.8.
a. What is the x-value that separates the bottom 65%?
b. A very special and lucrative scholarship goes to the students that score in the top 3%. What
is the least that score can be to receive that scholarship?
c. What are the x-values that are between the bottom 50% and the top 25%?

Answer 1

The Eleanor score is 2.06 deviations above the mean.

The Gerald score is 1.08 deviations above the mean.

Step-by-step explanation:

Previous concepts

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:

Where and

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

And if we replace we got:

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:

Where and

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

And if we replace we got:

So the Gerald score is 1.08 deviations above the mean.

Explanation:

The Eleanor score is 2.06 deviations above the mean.

The Gerald score is 1.08 deviations above the mean.

Step-by-step explanation:

Previous concepts

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:

Where and

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

And if we replace we got:

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:

Where and

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

And if we replace we got:

So the Gerald score is 1.08 deviations above the mean.