Question

According to the college board, scores by women on the SAT-I test were normally distributed with a mean of 998 and standard deviation of 202. Score by women on the ACT test are normally distributed with a mean of 20.9 and a standard deviation of 4.6. Assume that the two tests use different scales to measure the same aptitude

a) If a woman gets an SAT-I score in the 67thh percentile, find her actual SAT-I score and her equivalent ACT score

b) If a woman gets an SAT score of 1220, find her equivalent ACT score

Answer 1

(A) Using the z score and percentile table, z score corresponding to 67th percentile is 0.9741

Using the formula

FOR SAT-I exam, we have

setting the values, we get

0.9741 = (x-998)/202

by cross multiplication, we get

0.9741*202=x - 998

196.7682 = x-998

adding 998 on each side, we get

x = 998 +196.7682 = 1194.77 (rounded to 2 decimals)

FOR ACT exam, we have

setting the values, we get

0.9741 = (x-20.9)/4.6

by cross multiplication, we get

0.9741*4.6=x - 20.9

4.48086 = x-20.9

adding 20.9 on each side, we get

x = 20.9 + 4.48086 = 25.38 (rounded to 2 decimals)

(B) First, we need to find the z score corresponding to SAT score of 1220

we have and x = 1220

using the formula for z, we can write

z = (1220-998)/202 = 222/202 = 1.099

Calculation for ACT score

Now, we have z = 1.099 and

setting the values in the z score formula, we get

1.099 = (x-20.9)/4.6

by cross multiplication, we get

1.099*4.6=x - 20.9

5.0554 = x-20.9

adding 20.9 on each side, we get

x = 20.9 + 5.0554 = 25.96 (rounded to 2 decimals)