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There are two major tests of readiness for college, the ACT and the SAT. ACT scores are reported on a scale from 1 to 36. The distribution of ACT scores for more than 1 million students in a recent high school graduating class was roughly normal with mean μ = 20.8 and standard deviation σ = 4.8. SAT scores are reported on a scale from 400 to 1600. The SAT scores for 1.4 million students in the same graduating class were roughly normal with mean μ = 1026 and standard deviation σ = 209. How well must Abigail do on the SAT in order to place in the top 8% of all students? (Round your answer to the nearest whole number.)

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Answer:

1321.

Step-by-step explanation:

We have been given that SAT scores are reported on a scale from 400 to 1600. The SAT scores for 1.4 million students in the same graduating class were roughly normal with mean μ = 1026 and standard deviation σ = 209. We are asked to find the score that Abigail must get on the SAT in order to place in the top 8% of all students.

Top 8% is equal to 92% or more.

Let us find z-score corresponding to 92% or 0.92. Using normal distribution table, we get a z-score equal to 1.41.

Now, we will z-score formula to solve for the score as:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

[tex]1.41=\frac{x-1026}{209}[/tex]

[tex]1.41*209=\frac{x-1026}{209}*209[/tex]

[tex]294.69=x-1026[/tex]

[tex]294.69+1026=x-1026+1026[/tex]

[tex]1320.69=x[/tex]

[tex]x\approx 1321[/tex]

Therefore, Abigail must get a score of 1321 on the SAT in order to place in the top 8% of all students.