You buy a 903 acre farm for 3,642,850. What is the price per acre?

Ronald scores 700 on the math section of the SAT exam. The distribution of SAT scores is approximately normal with a mean of 500

Artist 52 [7]

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:

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

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