Question

For college-bound high school seniors in 1996, the nationwide mean SAT verbal score was 505 with a standard deviation of about 110, and the mean SAT math score was 508 with a standard deviation of about 110. Students who do well on the verbal portion of the SAT tend to do well on the mathematics portion. If the two scores for each student are added, the mean of the combined scores is 1,013. What is the standard deviation of the combined verbal and math scores?

a. 110 √2 (approximately 77.78)
b. 110
C. √1102+1102
d. 220
e. The standard deviation cannot be computed from the information given.

Answer 1

The standard deviation of the combined verbal and math scores, using the sum of variances formula, is
`\(110 √(2)\)` approximately 77.78. Hence, the correct choice is (a) 110 √2.

Given information:

Mean SAT verbal score
`(\(X\))` = 505

Standard deviation of SAT verbal score
`(\(SD_X\))` = 110

Mean SAT math score
`(\(Y\))` = 508

Standard deviation of SAT math score
`(\(SD_Y\))` = 110

Mean of the combined scores
`(\(X + Y\))` = 1,013

Given the means:

`\[ \text{Mean of } X + \text{Mean of } Y = 1,013 \]`

[ 505 + 508 = 1,013 ]

The formula for the variance of the sum of two variables is the sum of their individual variances:

`\[ \text{Variance of } X + \text{Variance of } Y = \text{Variance of } X + \text{Variance of } Y \]`

`\[ \text{Variance of } X + \text{Variance of } Y = SD_X^2 + SD_Y^2 = 110^2 + 110^2 = 2 * 110^2 \]`

The standard deviation is the square root of the variance:

`\[ \text{Standard deviation of } X + \text{Standard deviation of } Y = √(2 * 110^2) = 110 √(2) \]`

Therefore, the standard deviation of the combined verbal and math scores is approximately
`\(110 √(2)\)` , which is approximately 77.78. Thus, the answer is option (a).