Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.


Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.


a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.

Verified step by step guidance

1. Identify the key components of the problem: The Empirical Rule applies to data that is approximately normally distributed. The rule states that approximately 68% of the data falls within one standard deviation (σ) of the mean (μ), 95% within two standard deviations, and 99.7% within three standard deviations. From Exercise 29, determine the mean (μ) and standard deviation (σ) of the sample data.
2. Determine the z-scores for the given speed range (63 mph to 71 mph). Use the formula for a z-score: $z=\frac{x-\mu }{\sigma }$, where x is the value, μ is the mean, and σ is the standard deviation.
3. Using the z-scores, calculate the proportion of the data that falls between the two z-scores. For a normal distribution, this can be done by consulting a z-table or using statistical software to find the cumulative probabilities for each z-score and subtracting the smaller probability from the larger one.
4. Multiply the proportion obtained in the previous step by the total number of vehicles in the sample (75) to estimate the number of vehicles whose speeds fall within the specified range.
5. Verify the result by ensuring the calculations align with the Empirical Rule's expectations for a normal distribution. For example, check if the range corresponds to approximately one standard deviation from the mean, which would include about 68% of the data.

Key Concepts

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in estimating probabilities and understanding the spread of data in a normal distribution.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most observations cluster around the central peak (mean) and probabilities for values further away from the mean taper off equally in both directions. Understanding this distribution is crucial for applying the Empirical Rule effectively.

Sample Statistics

Sample statistics are numerical values calculated from a subset of a population, which are used to estimate population parameters. In this context, knowing the sample mean and standard deviation is essential for applying the Empirical Rule to estimate the number of vehicles within a specified speed range.