Travel Time to Work The frequency distribution listed in the table represents the travel time to work (in minutes) for a random sample of 895 U.S. adults.
a. Approximate the mean travel time to work for U.S. adults.

Verified step by step guidance

1. Identify the midpoint of each travel time interval. The midpoint is found by averaging the lower and upper bounds of each interval. For example, for the interval 0–9, the midpoint is \(\frac{0 + 9}{2} = 4.5\) minutes.
2. Multiply each midpoint by the corresponding frequency to find the total travel time contribution for each interval. This means calculating \(\text{midpoint} \times \text{frequency}\) for each row.
3. Sum all the products obtained in the previous step to get the total combined travel time for all 895 adults.
4. Sum all the frequencies to confirm the total number of observations, which should be 895 in this case.
5. Calculate the approximate mean travel time by dividing the total combined travel time by the total number of observations using the formula: \(\text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}\).

Key Concepts

Grouped Frequency Distribution

A grouped frequency distribution organizes data into intervals (or classes) and shows the number of observations within each interval. It helps summarize large data sets and is essential for estimating 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 is used as a representative value for all data points in that interval when calculating approximate statistics such as the mean.

Approximate Mean from Grouped Data

To approximate the mean from grouped data, multiply each class midpoint by its frequency, sum these products, and divide by the total number of observations. This weighted average provides an estimate of the central tendency when raw data is grouped.