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Bill Board is "lording" his SAT score over his friend, Rhoda Dendron, who took the ACT. "You only got a 25 in math," he chortled, "while I got a 300 in math." Given that the SAT has a μ of 500 and a σ of 100, and the ACT has a μ of 20 and a σ of 5, what is wrong with Bill’s logic (give the answer in both z scores and percentile ranks).

Sagot :

Answer:

Rhoda, whose ACT has a z-score of 1, scored in the 84th percentile in the ACT, compared to Bill, whose SAT has a z-score of -2, who scored in the 2nd percentile on the SAT. Due to the higher percentile(higher z-score)  on her test, Rhoda did better on her respective test than Bill, and thus, his logic is wrong.

Step-by-step explanation:

Z-score:

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

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

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 p-value, we get the probability that the value of the measure is greater than X.

Bill:

Scored 300, so [tex]X = 300[/tex]

SAT has a μ of 500 and a σ of 100.

His z-score is:

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

[tex]Z = \frac{300 - 500}{100}[/tex]

[tex]Z = -2[/tex]

[tex]Z = -2[/tex] has a p-value of 0.02.

This means that Bill, whose SAT score has Z = -2, scored in the 2nd percentile.

Rhoda

Scored 25, so [tex]X = 25[/tex].

ACT has a μ of 20 and a σ of 5

Her z-score:

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

[tex]Z = \frac{25 - 20}{5}[/tex]

[tex]Z = 1[/tex]

[tex]Z = 1[/tex] has a p-value of 0.84

This means that Rhoda, whose ACT score has Z = 1, scored in the 84th percentile.

What is wrong with Bill’s logic ?

Rhoda, whose ACT has a z-score of 1, scored in the 84th percentile in the ACT, compared to Bill, whose SAT has a z-score of -2, who scored in the 2nd percentile on the SAT. Due to the higher percentile(higher z-score)  on her test, Rhoda did better on her respective test than Bill, and thus, his logic is wrong.

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