The ACT and SAT are two college entrance exams. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1. The composite score on the SAT is approximately normally distributed with mean 1026 and standard deviation 210. Suppose you scored 26 on the ACT and 1240 on the SAT. Which exam did you score better on? Justify your reasoning using the normal model.

Verified step by step guidance

1. Identify the distributions for both exams: ACT scores are normally distributed with mean \( \mu_{ACT} = 21.1 \) and standard deviation \( \sigma_{ACT} = 5.1 \), and SAT scores are normally distributed with mean \( \mu_{SAT} = 1026 \) and standard deviation \( \sigma_{SAT} = 210 \).
2. Calculate the z-score for the ACT score of 26 using the formula: \[ z_{ACT} = \frac{X_{ACT} - \mu_{ACT}}{\sigma_{ACT}} = \frac{26 - 21.1}{5.1} \]
3. Calculate the z-score for the SAT score of 1240 using the formula: \[ z_{SAT} = \frac{X_{SAT} - \mu_{SAT}}{\sigma_{SAT}} = \frac{1240 - 1026}{210} \]
4. Interpret the z-scores: A higher z-score indicates a score further above the mean relative to the distribution's spread, meaning a better performance compared to other test takers.
5. Compare the two z-scores to determine on which exam you scored better. The exam with the higher z-score is the one where your score is relatively better.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by its mean and standard deviation, which determine the center and spread of the data. Many natural phenomena, including test scores, approximate this distribution, allowing us to use it for probability and comparison.

Z-Score

A z-score measures how many standard deviations a data point is from the mean. It standardizes different data points, enabling comparison across different scales or distributions. Calculated as (score - mean) / standard deviation, it helps determine relative performance on different exams.

Comparing Scores Using the Normal Model

To compare scores from different tests with different scales, convert each score to its z-score. The higher z-score indicates a better relative performance compared to the test-taking population. This method uses the normal model to justify which score is better standardized.