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Sagot :
Answer:
The actual SAT score is 2024.
The equivalent ACT score is 29.49.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
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.
SAT score that is in the 95th percentile
Scores on the SAT test are normally distributed with a mean of 1511 and a standard deviation of 312, which means that [tex]\mu = 1511, \sigma = 312[/tex]
95th percentile is X when Z has a pvalue of 0.95, so X when Z = 1.645. The score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 1511}{312}[/tex]
[tex]X - 1511 = 1.645*312[/tex]
[tex]X = 2024[/tex]
The actual SAT score is 2024.
Equivalent ACT score:
The equivalent ACT score is the 95th percentile of ACT scores.
Scores on the ACT test are normally distributed with a mean of 21.1 and a standard deviation of 5.1, which means that [tex]\mu = 21.1, \sigma = 5.1[/tex]. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 21.1}{5.1}[/tex]
[tex]X - 21.1 = 1.645*5.1[/tex]
[tex]X = 29.49[/tex]
The equivalent ACT score is 29.49.
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