Question

In a certain year, when she was a high school senior, Idonna scored 620 on the mathematics part of the SAT. The distribution of SAT math scores in that year was Normal with mean 512 and standard deviation 117. Jonathan took the ACT and scored 24 on the mathematics portion. ACT math scores for that year were Normally distributed with mean 20.5 and standard deviation 5.4.
Find the standardized scores (±0.01) for both students. Assuming that both tests measure the same kind of ability, who had the higher score?
Idonna's standardized score is ____
Jonathan's standardized score is _____

a. Scores are equal
b. Idonna's score is less than Jonathan's
c. Idonna's score is higher than Jonathan's

Answer 1

Answer:

Idonna's standardized score is 0.92.

Jonathan's standardized score is 0.65.

Assuming that both tests measure the same kind of ability, who had the higher score?

c. Idonna's score is higher than Jonathan's

Step-by-step explanation:

Problems of normally distributed samples can be solved using 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 problem, we have that:

Idonna's standardized score is:

Idonna scored 620 on the mathematics part of the SAT. The distribution of SAT math scores in that year was Normal with mean 512 and standard deviation 117. So $X = 620, \mu = 512, \sigma = 117$

Her standardized score is Z

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

$Z = \frac{620 - 512}{117}$

$Z = 0.92$

Idonna's standardized score is 0.92.

Jonathan's standardized score is:

Jonathan took the ACT and scored 24 on the mathematics portion. ACT math scores for that year were Normally distributed with mean 20.5 and standard deviation 5.4. So $X = 24, \mu = 20.5, \sigma = 5.4$

His standardized score is Z

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

$Z = \frac{24 - 20.5}{5.4}$

$Z = 0.65$

Jonathan's standardized score is 0.65.

Assuming that both tests measure the same kind of ability, who had the higher score?

Idonna's had the higher z-score, so her score is higher.

So the correct answer is:

c. Idonna's score is higher than Jonathan's