Question

If the SAT follows N(998,202), answer the following questions:

a. P( 885 < x < 1220? Show your work including the shading of normal distribution plot.

b. At what score do 45% of the scores exceed that score? Show your work including the shading of the normal distribution plot.

c. P(z > 1100)? Show your work including the shading of the normal distribution plot.

Answer 1

Let random variable X be SAT scores.

X follows normal distribution with mean = = 998 and variance = = 202

Standard deviation = = 14.2127              (Round to 4 decimal)

a) Here we have to find P(885 < X < 1220)

.

                                           

                                                   (Where z is standard normal variable)

                                           = P(z < 15.62) - P(z < -7.95)

(From statistical table of z values, z values greater than 3.4, use 1 to approximate the area and for values of z less than -3.4, use 0 to approximate the area.)

                                           = 1 - 0

                                           = 1

P(885 < X < 1220) = 1

b) Here we have to find x such that P(X > x) = 0.45

1 - P(X < x) = 0.45

P(X < x) = 0.55

z for area = 0.55 is 0.13                     (From statistical table)

    = 998 + 0.13 * 14.2127

    = 998 + 1.847651

    = 999.8477

45% of the scores exceed the score 999.8477

c)

P(z > 1100) = 1 - P(z < 1100)      (where z is standard normal variable)

                   = 1 - 1                  (From statistical table of z values)

                   = 0