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3. Scores on the SAT form a normal distribution with a mean of 500 and StandardDeviation of 100.a) What is the minimum score necessary to be in the top 10% of the SATdistribution?b) Find the range of values that defines the middle 80% of the distribution ofSAT scores (372 and 628). (The z scores)

Answer :

Given

Mean is 500 and Standard Deviation is 100.

Part A

Using Z- Score

The minimum score necessary to be in the top 10% of the SAT distribution means 90% and above

[tex]\begin{gathered} Z=\frac{x-\mu}{\sigma} \\ \\ x=0.9 \\ \mu=500 \\ \sigma=100 \\ 1.282=\frac{x-500}{100} \\ \\ x=628.2 \end{gathered}[/tex]

Thus, the minimum score is 628. To the nearest whole number.

Part B

We want to determine the range of values that correspond to the probability;

[tex]Pr(-ZThe z scores corresponding to this is; [tex]z=+1.282\text{ }&-1.282[/tex]

Thus;

[tex]\begin{gathered} 1.282=\frac{x-500}{100}\text{ and }-1.282=\frac{x-500}{100} \\ 628.2=x\text{ }and\text{ }371.8 \end{gathered}[/tex]

Thus, the range of scores are 372 and 628

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