Question

In 2017 national results for the SAT test show that for college-bound seniors the average total score — sum of the Math and Evidence-Based Reading and Writing sub-sections — was 1060 and the standard deviation was 195. Also in 2017, national results for the ACT test show that for college-bound seniors the average composite ACT score was 21.0 and the standard deviation was 5.4. Assume both the total SAT scores and composite ACT scores follow a normal distribution. More info: SAT Score Structure,

C1) What is the cut-off composite ACT score for being considered in the top 15% of all college-bound seniors? (2 pts)

For the top 15% of all college bound seniors, the cut off composite act score being considered is

C2) What is the cut-off total SAT score for being considered in the top 15% of all college-bound seniors? (2 pts)

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C3) Suppose a college-bound senior only took the SAT and her composite score was 1440. What might we expect her composite ACT score to be? (2 pts)

C4) What is the probability that a randomly selected college-bound senior scored between 1250 and 1550 on the combined SAT? (2 pts)

C5) What proportion of college-bound seniors had a composite ACT score below 10 or above 30? (2 pts)

C6) Suppose Chandler had a composite ACT score of 26 and Joey had a total SAT score of 1260. Whose score would you consider more impressive? Justify your answer. (2 pts)

C7) Suppose we randomly sampled 30 college-bound seniors.

1. What is the probability that the mean composite ACT score for the sample is below 18? (2 pts)

2. What is the probability that the mean total SAT score for the sample is above 1100? (2 pts)

Answer 1

In 2017 national results for the SAT test show that for college-bound seniors the average total score — sum of the Math and Evidence-Based Reading and Writing sub-sections — was 1060 and the standard deviation was 195. Also in 2017, national results for the ACT test show that for college-bound seniors the average composite ACT score was 21.0 and the standard deviation was 5.4. Assume both the total SAT scores and composite ACT scores follow a normal distribution. More info: SAT Score Structure,

C1) What is the cut-off composite ACT score for being considered in the top 15% of all college-bound seniors? (2 pts)

Z value for top 15% = 1.036

X=mean+z*sd

X=1060+1.036*195 =1262.02

For the top 15% of all college bound seniors, the cut off composite act score being considered is 1262.02

C2) What is the cut-off total SAT score for being considered in the top 15% of all college-bound seniors? (2 pts)

X=21+1.036*5.4=26.5944

C3) Suppose a college-bound senior only took the SAT and her composite score was 1440. What might we expect her composite ACT score to be? (2 pts)

Z value for 1440, z =(1440-1060)/195 = 1.95

Required score = 21+1.95*5.4 =31.53

C4) What is the probability that a randomly selected college-bound senior scored between 1250 and 1550 on the combined SAT? (2 pts)

Z value for 1250, z =(1250-1060)/195 = 0.97

Z value for 1550, z =(1550-1060)/195 = 2.51

P( 1250<x<1550) = P( 0.97<z<2.51)

=P( z < 2.51)-P(z <0.97)

=0.994-0.834

=0.16

C5) What proportion of college-bound seniors had a composite ACT score below 10 or above 30? (2 pts)

Z value for 10, z =(10-21)/5.4 = -2.04

Z value for 30, z =(30-21)/5.4 = 1.67

P( 10<x<30) = P( -2.04<z<1.67)

=P( z < 1.67)-P(z <-2.04)

= 0.9525- 0.0207

₌0.9318

C6) Suppose Chandler had a composite ACT score of 26 and Joey had a total SAT score of 1260. Whose score would you consider more impressive? Justify your answer. (2 pts)

Z value for 1260, z =(1260-1060)/195 = 1.03

Z value for 26, z =(26-21)/5.4 = 0.93

Since z value for total SAT score of 1260 is large , it is more impressive.

C7) Suppose we randomly sampled 30 college-bound seniors.

1. What is the probability that the mean composite ACT score for the sample is below 18? (2 pts)

Standard error =sd/sqrt(n) =5.4/sqrt(30) =0.9859

Z value for 18, z =(18-21)/0.9859 = -3.04

P( mean x <18) =P( z < -3.04)

=0.0012

2. What is the probability that the mean total SAT score for the sample is above 1100? (2 pts)

Standard error =sd/sqrt(n) =195/sqrt(30) =35.602

Z value for 1100, z =(1100-1060)/ 35.602 =1.12

P( mean x >1100) = P( z >1.12)

=0.1314