Question

4. National SAT (Scholastic Aptitude Test) scores for high school students in the U.S.A. are normally distributed with a mean of 500 and a standard deviation of 116. What is the percentage of students that score (a) above 700? (C) between 650 and 800? (b) under 400? (d) within 50 of the mean?

Answer 1

Solution :

Given that ,

mean = $\mu$ = 500

standard deviation = $\sigma$ = 116

(a)

P(x > 700) = 1 - P(x < 700)

= 1 - P[(x - $\mu$ ) / $\sigma$ < (700 - 500) / 116]

= 1 - P(z < 1.72)

= 1 - 0.9573

= 0.0427

Percentage = 4.27%

(b)

P(x < 400) = P[(x - $\mu$ ) / $\sigma$ < (400 - 500) / 116]

= P(z < -0.86)  

= 0.1949

Percentage = 19.49%

(c)

P(650 < x < 800) = P[(650 - 500)/ 116) < (x - $\mu$ ) / $\sigma$   < (800 - 500) / 116) ]

= P(1.29 < z < 2.59)

= P(z < 2.59) - P(z < 1.29)

= 0.9952 - 0.9015

= 0.0937

Percentage = 9.37%

(d)

P(450 < x < 550) = P[(450 - 500)/ 116) < (x - $\mu$ ) / $\sigma$   < (550 - 500) / 116) ]

= P(-0.43 < z < 0.43)

= P(z < 0.43) - P(z < -0.43)

= 0.6664 - 0.3336

= 0.3328

Percentage = 33.28%