The following data represent the length of time (in minutes) between eruptions of Old Faithful in Yellowstone National Park.
a. Approximate the mean length of time between eruptions. .

Verified step by step guidance

1. Step 1: Identify the midpoints of each class interval. The midpoint is found by averaging the lower and upper bounds of each interval. For example, for the interval 40-49, the midpoint is calculated as \(\frac{40 + 49}{2}\).
2. Step 2: Multiply each midpoint by its corresponding frequency to find the total contribution of each class to the overall sum. This means calculating \(\text{midpoint} \times \text{frequency}\) for each class.
3. Step 3: Sum all the products obtained in Step 2 to get the total weighted sum of all time intervals.
4. Step 4: Sum all the frequencies to find the total number of observations.
5. Step 5: Calculate the approximate mean length of time between eruptions by dividing the total weighted sum by the total frequency using the formula: \(\text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}\).

Key Concepts

Grouped Data and Class Intervals

Grouped data is data organized into intervals or classes, each with a frequency indicating how many observations fall within that range. Understanding class intervals helps in summarizing large data sets and is essential for calculating measures like the mean when individual data points are unavailable.

Midpoint of Class Intervals

The midpoint of a class interval is the value halfway between the lower and upper boundaries of the interval. It serves as a representative value for all data points in that class, allowing approximation of statistics such as the mean when only grouped data is available.

Calculating the Mean from Grouped Data

To approximate the mean from grouped data, multiply each class midpoint by its frequency, sum these products, and then divide by the total number of observations. This weighted average provides an estimate of the central tendency when raw data points are not accessible.