The Empirical Rule SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114.


Source: College Board


b. What percentage of SAT scores is less than 401 or greater than 629?

Verified step by step guidance

1. Identify the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the SAT Math scores: \( \mu = 515 \) and \( \sigma = 114 \).
2. Calculate the z-scores for the values 401 and 629 using the formula for a z-score: \[ \text{z} = \frac{X - \mu}{\sigma} \] where \( X \) is the value of interest.
3. Interpret the z-scores in terms of the Empirical Rule, which states that for a bell-shaped distribution: - About 68% of data falls within 1 standard deviation of the mean, - About 95% within 2 standard deviations, - About 99.7% within 3 standard deviations.
4. Determine the percentage of scores between 401 and 629 by finding the proportion of data within the corresponding z-score range using the Empirical Rule.
5. Subtract the percentage found in the previous step from 100% to find the percentage of SAT scores that are less than 401 or greater than 629.

Key Concepts

Empirical Rule

The Empirical Rule states that for a bell-shaped (normal) distribution, about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This helps estimate the percentage of values in certain intervals without exact calculations.

Standard Deviation

Standard deviation measures the average distance of data points from the mean, indicating the spread or variability in the data. In this problem, it helps define intervals around the mean to apply the Empirical Rule.

Complement Rule in Probability

The Complement Rule states that the probability of an event occurring is 1 minus the probability of it not occurring. Here, to find the percentage of scores less than 401 or greater than 629, we use complements of the middle range percentages given by the Empirical Rule.