In some state the mean ACT score in Mathematics is mu_ACT = 20 and the standard

Question

In some state the mean ACT score in Mathematics is mu_ACT = 20 and the standard deviation is sigma_ACT = 5, while the mean SAT score in Mathematics is mu_SAT = 480 and the standard deviation is sigma_SAT = 100. suppose that the ACT scores and the SAT scores have a normal distribution. Terence gets a score of 12 on the Mathematics portion of the ACT and Philip gets a score of 480 on the Mathematics portion of the SAT. Answer questions (a)-(d) using the Empirical Rule. You should not use R here since the point of this exercise is to practice the Empirical Rule. (a) what proportion of ACT scores are less than 15? (b) what proportion of SAT scores are greater than 680? (c) What proportion of ACT scores are between 10 and 25? (d) What proportion of SAT scores are between 480 and 680? (e) What is the z-score of Terence? (f) What is the z-score of Philip? (g) who did better in Mathematics, Terence or Philip? Terence Philip

Explanation / Answer

ACT

mean (u1)= 20

standard deviation (s1)=5

USAT

mean (u2) = 480

standard deviation (s2) = 100

a) Proporition of ACT scores < 15

p(X<15)

Z = (X-u1)/s1

= (15-20)/5 = -1

so, p(X<15) = p(Z<-1) = 0.1587 (from Z table)

b) p(X>680)

Z = (X-u2)/s2

= (680-480)/100

= 2

p(X>680)=p(Z>2) = (1-0.9772) = 0.0228

c) p(10<X<25)

Similar conversion to Z scale as above

p(-2<Z<1) = 0.8413 - 0.0228 = 0.8185

d) p(480<x<680)

= p(0<Z<2)

= 0.9772-0.5 = 0.4772

e)Terrace, Z = (12 - 20)/ 5 = -1.6

f) Phillips, Z = (480-480)/100 = 0

g) Phillips is better having better Z score

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