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Bad White [126]
3 years ago
8

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).
Mathematics
1 answer:
SCORPION-xisa [38]3 years ago
4 0

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 \mu and standard deviation \sigma, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Bill:

Scored 300, so X = 300

SAT has a μ of 500 and a σ of 100.

His z-score is:

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

Z = \frac{300 - 500}{100}

Z = -2

Z = -2 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 X = 25.

ACT has a μ of 20 and a σ of 5

Her z-score:

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

Z = \frac{25 - 20}{5}

Z = 1

Z = 1 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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