Question

each year thousands of high school students take either the sat or the act, standardized tests used in the college admissions process. combined sat math and verbal scores go as high as 1600, while the maximum act composite score is 36. since the two exams use very different scales, comparisons of performance are difficult. a convenient rule of thumb is sat = 40×act + 150; that is, multiply an act score by 40 and add 150 points to estimate the equivalent sat score. an admissions officer reported the following statistics about the act scores of 2328 students who applied to her college one year. find the summaries of equivalent sat scores.
lowest score = 17 mean = 28 standard deviation = 2
q3 = 31 median = 29 iqr = 4
lowest score for the sat = 830
mean for the sat = 1270
standard deviation for the sat = 2

Explanation:

Step1: Calculate lowest SAT score

Use the formula SAT = 40×ACT + 150. Substitute ACT = 17.
SAT = 40×17+150 = 680 + 150 = 830

Step2: Calculate mean SAT score

Substitute the mean ACT score of 28 into the formula.
SAT = 40×28+150 = 1120+150 = 1270

Step3: Calculate standard - deviation of SAT scores

If \(y = a x + b\) (where \(y\) is SAT, \(x\) is ACT, \(a = 40\) and \(b = 150\)), the standard - deviation of \(y\) is \(SD_y=a\times SD_x\). Given \(SD_x = 2\) and \(a = 40\), then \(SD_y=40\times2 = 80\)

Step4: Calculate Q3 for SAT scores

Substitute ACT Q3 value of 31 into the formula. SAT = 40×31+150 = 1240+150 = 1390

Step5: Calculate median for SAT scores

Substitute ACT median value of 29 into the formula. SAT = 40×29+150 = 1160+150 = 1310

Step6: Calculate IQR for SAT scores

Since \(IQR_y=a\times IQR_x\) and \(a = 40\), \(IQR_x = 4\), then \(IQR_y=40\times4 = 160\)

Answer:

Lowest score for the SAT = 830
Mean for the SAT = 1270
Standard deviation for the SAT = 80
Q3 for the SAT = 1390
Median for the SAT = 1310
IQR for the SAT = 160