"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


a. What percentage of SAT scores is between 401 and 629?"

Verified step by step guidance

1. Identify the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the SAT Math scores. Here, \( \mu = 515 \) and \( \sigma = 114 \).
2. Calculate how many standard deviations away from the mean the values 401 and 629 are by using the z-score formula: \[ z = \frac{X - \mu}{\sigma} \] where \( X \) is the value of interest.
3. Compute the z-scores for 401 and 629: \[ z_{401} = \frac{401 - 515}{114} \] \[ z_{629} = \frac{629 - 515}{114} \]
4. Use the Empirical Rule (68-95-99.7 rule) which states that approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean. Determine where the z-scores fall relative to these intervals.
5. Based on the z-scores and the Empirical Rule, estimate the percentage of SAT scores that lie between 401 and 629 by identifying the corresponding percentage range between those z-scores.

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 rule helps estimate the percentage of values within a certain range without exact calculations.

Mean and Standard Deviation

The mean is the average value of a data set, representing its center, while the standard deviation measures the spread or variability around the mean. In a normal distribution, these parameters define the shape and scale of the data.

Interpreting Data Ranges in Normal Distributions

To find the percentage of data between two values in a normal distribution, determine how many standard deviations each value is from the mean, then use the Empirical Rule or standard normal tables to estimate the proportion of data within that range.