Final answer:
For data sets with a bell-shaped distribution, the Empirical Rule states that about 68% of the data falls within one standard deviation of the mean, making it the correct answer to the student's question.
Step-by-step explanation:
For data sets having a distribution that is approximately bell-shaped, the Empirical Rule states that about 68% of all data values fall within one standard deviation from the mean. This rule is a key concept in statistics and is often introduced in high school mathematics when students learn about the properties of the normal distribution. This rule states:
- Approximately 68 percent of the data falls within one standard deviation of the mean.
- Approximately 95 percent of the data falls within two standard deviations of the mean.
- More than 99 percent of the data falls within three standard deviations of the mean.
Chebyshev's Theorem, in contrast, can be applied to any distribution, not just bell-shaped distributions, and states that a minimum percentage of data lies within a certain number of standard deviations from the mean, although these percentages are less specific than those given by the Empirical Rule for normal distributions.
The Empirical Rule is also sometimes referred to as the 68-95-99.7 rule, reflecting the percentages of data within one, two, and three standard deviations of the mean, respectively.
To calculate whether a sample mean is within a certain number of standard deviations from the population mean, we can make use of the formula for the standard deviation of a sample and apply the rule to estimate the distribution of sample data about the mean.