- If your score on your next statistics test is converted to a \( z \) score, which of these \( z \) scores would you prefer: \( -2.00,-1.00,0,1.00,2.00 \) ? Why?

I would prefer a \( z \)-score of **2.00**. Here's why: ### Understanding \( z \)-Scores A \( z \)-score represents how many standard deviations a particular score is from the **mean** (average) of a dataset. The formula for calculating a \( z \)-score is: \[ z = \frac{(X - \mu)}{\sigma} \] Where: - \( X \) = your score - \( \mu \) = mean of all scores - \( \sigma \) = standard deviation ### Interpretation of Given \( z \)-Scores - **-2.00**: Your score is 2 standard deviations *below* the mean. - **-1.00**: Your score is 1 standard deviation *below* the mean. - **0**: Your score is exactly at the mean. - **1.00**: Your score is 1 standard deviation *above* the mean. - **2.00**: Your score is 2 standard deviations *above* the mean. ### Why Prefer a \( z \)-Score of 2.00? - **Higher Performance**: A \( z \)-score of 2.00 indicates that your score is significantly above average. Specifically, you're two standard deviations above the mean, placing you in approximately the top 2.5% of all scores (assuming a normal distribution). - **Competitive Advantage**: Achieving a higher \( z \)-score can be beneficial for academic recognition, scholarships, or other opportunities that consider performance metrics. - **Confidence in Understanding**: A higher \( z \)-score may reflect a stronger grasp of the material, which can be advantageous for future courses and applications. ### Visual Representation Imagine the distribution of test scores as a bell curve: ``` <----|----|----|----|----> -2 -1 0 1 2 ``` - **Left Side (-2 to 0)**: Below average - **Center (0)**: Average - **Right Side (0 to 2)**: Above average Being on the far right side means excelling compared to peers. ### Conclusion Choosing a \( z \)-score of **2.00** signifies outstanding performance, placing you well above the average student. This not only reflects well academically but can also open doors to various opportunities that reward high-achieving individuals.