A nationwide standardized exam consists of a multiple-choice section and a free
ID: 3131075 • Letter: A
Question
A nationwide standardized exam consists of a multiple-choice section and a free response section. For each section, the mean and standard deviation are reported to be as shown in the following table.
Let's define x1 and x2 as the multiple-choice score and the free-response score, respectively, of a student selected at random from those taking the exam. We are also interested in the variable y = overall exam score.
Suppose that the free-response score is given twice the weight of the multiple-choice score in determining the overall exam score. Then the overall score is computed as
y = x1 + 2x2.
What are the mean and standard deviation of y?
Because y = x1 + 2x2 is ---Select--- a linear a constant a random an average combination of x1 and x2, the mean of y is computed as follows.
What about the variance and standard deviation of y? To use the rule for variance of ---Select--- random constant average linear combinations, x1 and x2 must be ---Select--- independent equal random dependent unequal . It is unlikely that the value of x1 (a student's multiple choice score) would be unrelated to the value of x2 (the same student's free response score), because it seems probable that students who score well on one section of the exam will also tend to score well on the other section. Therefore, it would not be appropriate to use the formulas mentioned above to calculate the variance and standard deviation.
Mean Standard Deviation Multiple Choice 45 6 Free Response 42 8Explanation / Answer
Mean = 45 + (2 × 42) = 129
It is unlikely that the value of x1 (a student's multiple choice score) would be unrelated to the value of x2 (the same student's free response score), because it seems probable that students who score well on one section of the exam will also tend to score well on the other section. Therefore, it would not be appropriate to use the formulas mentioned above to calculate the variance and standard deviation.
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