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Use the normal distribution of SAT critical reading scores for which the mean is 504 and the standard deviation is 112. Assume the variable x is normally distributed.
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(a) What percent of the SAT verbal scores are less than 675​?
(b) If 1000 SAT verbal scores are randomly​ selected, about how many would you expect to be greater than 525​?

Approximately
_____% of the SAT verbal scores are less than 675.
​(Round to two decimal places as​ needed.)

Respuesta :

LammettHash

a. Transform [tex]X[/tex], the random variable representing SAT scores, to [tex]Z[/tex], the same random variable but following the standard normal distribution, via

[tex]X=\mu_XZ+\sigma_X[/tex]

[tex]P(X<675)=P\left(\dfrac{X-504}{112}<\dfrac{675-504}{112}\right)\approxP(Z<1.5268)\approx0.9366[/tex]

so that about 93.77% of scores fall below 675.

b. Use the same strategy to compute [tex]P(X>525)[/tex]:

[tex]P(X>525)=P\left(\dfrac{X-504}{112}>\dfrac{525-504}{112}\right)\approx P(Z>0.1875)\approx0.5744[/tex]

Then out of 1000 scores, you would expect about 574 of them to exceed 525.

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