Question

Eleanor scores 680 on the mathematics part of the SAT. The distribution of SAT math scores in recent years has been Normal with mean 561 and standard deviation 110. Gerald takes the ACT Assessment mathematics test and scores 27. ACT math scores are Normally distributed with mean 20.2 and standard deviation 2.1.

1. What is Elanor's standardized score?
2. What is Gerald's standardized score?
3. Assuming that both tests measure the same kind of ability, who has the higher score?
A. Gerald.
B. Elanor.
C. They both did equally well.

Answer 1

1. Elanor's standardized score is 1.08.

2. Gerald's standardized score is 3.24.

3. A. Gerald.

Explanation:

Z-score:

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

`Z = (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.

1. What is Elanor's standardized score?

Eleanor scores 680 on the mathematics part of the SAT. The distribution of SAT math scores in recent years has been Normal with mean 561 and standard deviation 110. This means that we have
`X = 680, \mu = 561, \sigma = 110`.

The standardized score is:

`Z = (X - \mu)/(\sigma)`

`Z = (680 - 561)/(110)`

`Z = 1.08`

Elanor's standardized score is 1.08.

2. What is Gerald's standardized score?

Gerald takes the ACT Assessment mathematics test and scores 27. ACT math scores are Normally distributed with mean 20.2 and standard deviation 2.1. This means that we have
`X = 27, \mu = 20.2, \sigma = 2.1`

`Z = (X - \mu)/(\sigma)`

`Z = (27 - 20.2)/(2.1)`

`Z = 3.24`

Gerald's standardized score is 3.24.

3. Assuming that both tests measure the same kind of ability, who has the higher score?

Due to the higher standardized score, Gerald