Explore: The Empirical Rule (for Normal Distributions) Solutions

The Empirical Rule (for Normal Distributions) Solutions

The Empirical Rule is also called the 68-95-99.7 Rule . 
For normally distributed data only, the Empirical Rule says: 
- The area to the right of the mean represents 50% of the data. 
- When given the mean (𝜇) and standard deviation (σ) of a population, you can approximate the area any single data will fall into by considering how far above/below it is from the mean. 
- The range of data can be written using either interval notation or a compound inequality.

Empirical Rule

Example 4

A snack bag company fills bags using a machine. The production line manager’s goal is for each filled bag to weigh 100 grams. Because the filling machine is very accurate, most bags are close to the target weight, with only a few bags being slightly under or slightly over the goal weight.

Name the mean and standard deviation.

$μ = 100, σ = 5.5$

How likely is it that a bag will weigh between 89 grams and 111 grams?

95%

If 250 bags were selected randomly from the production lines, how many would weigh less than 111 grams?

$89 - 111 = 95 % 100 - 95 % = 5 % \frac{5 %}{2} = 2.5 % 95 % + 2.5 % = 97.5 % 0.975 (250) = 243.75$

Approx. 244 bags will weigh < 111 grams

The manager notices that another filling machine has a standard deviation of 2.6, but has the same mean. Explain which machine should be used.

The machine with $σ = 2.6$ should be used because the weight of the snack bags will vary by a smaller number.

Example 5

A high school track coach compiled data over her coaching career, comparing pre-season and post-season mile times.

Name the mean and standard deviation for the pre-season.

$μ = 7 : 10, σ = 0 : 34$

Complete the post-season normal distribution when $μ = 6 : 25, σ = 0 : 20$ .

See graph.

Name the interval for 95% of the pre- and post-season mile times.

pre: $[5 : 52, 8 : 18]$

post: $[5 : 45, 7 : 05]$

Explain how the data shows that mile times have improved.

The 𝜇 in the post-season is lower, meaning the runners are faster. The 𝜎 is smaller, meaning there is less variance between the slowest and fastest runners.