Question

Ronald scores 700 on the math section of the SAT exam. The distribution of SAT scores is approximately normal with a mean of 500 and a standard deviation of 100. Rubin takes the ACT math exam and scores 31 on the math portion. ACT scores are approximately normally distributed with a mean of 18 and a standard deviation of 6.7
(a) Relative to their peers who also took the tests, who performed better on his test? Explain.
(b) A certain school will only consider those students who score in the top 1% in the math section. What grades would Ronald and Rubin have to receive on their respective tests to be considered for admission?
(c) Between what two grades does 95% of the population fall for the ACT and the SAT exams?

Answer 1

Answer:

a) Due to the higher z-score, Ronald performed better relative to his peers on the test.

b) Ronald needed a grade of at least 732.5, and Rubin of at least 33.58.

c) 95% of the population fall between graded of 4.868 and 31.132 on the ACT.

95% of the population fall between graded of 304 and 696 on the SAT.

Step-by-step explanation:

Normal probability distribution

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

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

$Z = \frac{X - \mu}{\sigma}$

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.

In this question:

(a) Relative to their peers who also took the tests, who performed better on his test? Explain.

We have to find whoever has the higher z-score.

Ronald:

Ronald scores 700 on the math section of the SAT exam. The distribution of SAT scores is approximately normal with a mean of 500 and a standard deviation of 100. So the z-score is found when $X = 700, \mu = 500, \sigma = 100$. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{700 - 500}{100}$

$Z = 2$

Rubin:

Rubin takes the ACT math exam and scores 31 on the math portion. ACT scores are approximately normally distributed with a mean of 18 and a standard deviation of 6.7. So the z-score is found when $X = 31, \mu = 18, \sigma = 6.7$. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{31 - 18}{6.7}$

$Z = 1.94$

Due to the higher z-score, Ronald performed better relative to his peers on the test.

(b) A certain school will only consider those students who score in the top 1% in the math section. What grades would Ronald and Rubin have to receive on their respective tests to be considered for admission?

They have to be in the 100 - 1 = 99th percentile, that is, they need a z-score with a pvalue of at least 0.99. So we need to find for them X when Z = 2.325.

Ronald:

$Z = \frac{X - \mu}{\sigma}$

$2.325 = \frac{X - 500}{100}$

$X - 500 = 232.5$

$X = 732.5$

Rubin:

$Z = \frac{X - \mu}{\sigma}$

$2.325 = \frac{X - 18}{6.7}$

$X - 18 = 15.58$

$X = 33.58$

Ronald needed a grade of at least 732.5, and Rubin of at least 33.58.

(c) Between what two grades does 95% of the population fall for the ACT and the SAT exams?

They fall between the 100 - (95/2) = 2.5th percentile and the 100 + (95/2) = 97.5th percentile, that is, they fall between X when Z = -1.96 and X when Z = 1.96.

ACT:

Lower bound:

$Z = \frac{X - \mu}{\sigma}$

$-1.96 = \frac{X - 18}{6.7}$

$X - 18 = -1.96*6.7$

$X = 4.868$

Upper bound:

$Z = \frac{X - \mu}{\sigma}$

$1.96 = \frac{X - 18}{6.7}$

$X - 18 = 1.96*6.7$

$X = 31.132$

95% of the population fall between graded of 4.868 and 31.132 on the ACT.

SAT:

Lower bound:

$Z = \frac{X - \mu}{\sigma}$

$-1.96 = \frac{X - 500}{100}$

$X - 500 = -196$

$X = 304$

Upper bound:

$Z = \frac{X - \mu}{\sigma}$

$1.96 = \frac{X - 500}{100}$

$X - 500 = 196$

$X = 696$

95% of the population fall between graded of 304 and 696 on the SAT.