## SAT Writing scores are normally distributed with a mean of 491 and a standard deviation of 113.A university plans to send letters of recogni

Question

SAT Writing scores are normally distributed with a mean of 491 and a standard deviation of 113.A university plans to send letters of recognition to students whose scores are in the top 8%. What is the minimum score required for a letter of recognition?

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2021-09-29T02:26:45+00:00
2021-09-29T02:26:45+00:00 1 Answer
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## Answers ( )

Answer:And if we solve for a we got

So the value of height that separates the bottom 92% of data from the top 8% is 649.765.

Step-by-step explanation:Previous concepts

Normal distribution, is a “probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean”.The

Z-scoreis “a numerical measurement used in statistics of a value’s relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean”.Solution to the problemLet X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:

Where and

For this part we want to find a value a, such that we satisfy this condition:

(a)

(b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.92 of the area on the left and 0.08 of the area on the right it’s z=1.405. On this case P(Z<1.405)=0.92 and P(z>0.92)=0.08

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

So the value of height that separates the bottom 92% of data from the top 8% is 649.765.