To understand how well Alexandra must score on Exam B to perform equivalently as she did on Exam A, we need to compare her performance on both exams using standardized scores, also known as z-scores. Here’s a detailed, step-by-step explanation of the process:
1. Calculate the Z-Score for Exam A:
- Z-score formula:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\(X\)[/tex] is the score, [tex]\(\mu\)[/tex] is the mean, and [tex]\(\sigma\)[/tex] is the standard deviation.
- For Exam A:
- Alexandra's score ([tex]\(X\)[/tex]) = 460
- Mean ([tex]\(\mu\)[/tex]) = 500
- Standard deviation ([tex]\(\sigma\)[/tex]) = 100
- Plug these values into the formula:
[tex]\[
Z_A = \frac{460 - 500}{100} = \frac{-40}{100} = -0.4
\][/tex]
2. Interpret the Z-Score for Exam A:
- The Z-score of -0.4 means that Alexandra scored 0.4 standard deviations below the mean on Exam A.
3. Calculate the Equivalent Score for Exam B:
- Alexandra needs to score the same number of standard deviations below the mean on Exam B.
- The equivalent Z-score for Exam B will be identical to the Z-score from Exam A, which is -0.4.
- We then use the Z-score to find the corresponding raw score for Exam B.
- Use the Z-score formula rearranged to solve for [tex]\(X\)[/tex]:
[tex]\[
X = Z \times \sigma + \mu
\][/tex]
- For Exam B:
- Z = -0.4
- Mean ([tex]\(\mu\)[/tex]) = 700
- Standard deviation ([tex]\(\sigma\)[/tex]) = 20
- Plug these values into the formula:
[tex]\[
X_B = -0.4 \times 20 + 700 = -8 + 700 = 692
\][/tex]
4. Conclusion:
- For Alexandra to perform equivalently on Exam B as she did on Exam A, she needs to score 692.
So, to do equivalently well on Exam B as she did on Exam A, Alexandra must score 692.
