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.

b. Approximate the standard deviation travel time to work for U.S. adults.

Verified step by step guidance

1. Step 1: Determine the midpoint for 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 \(\frac{0 + 9}{2} = 4.5\). Repeat this for all intervals.
2. Step 2: Multiply each midpoint by its corresponding frequency to find the total travel time contribution for each interval. This will help in calculating the mean travel time.
3. Step 3: Calculate the mean travel time \(\bar{x}\) using the formula: \(\bar{x} = \frac{\sum (f_i \times x_i)}{n}\) where \(f_i\) is the frequency for interval \(i\), \(x_i\) is the midpoint for interval \(i\), and \(n\) is the total number of observations (895 in this case).
4. Step 4: Calculate the squared deviation for each midpoint from the mean, i.e., \((x_i - \bar{x})^2\), then multiply each squared deviation by its corresponding frequency \(f_i\) to get \(f_i (x_i - \bar{x})^2\).
5. Step 5: Sum all the values from Step 4 and divide by the total number of observations minus 1 (i.e., \(n - 1\)) to find the sample variance: \(s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{n - 1}\) Finally, take the square root of the variance to get the standard deviation: \(s = \sqrt{s^2}\).

Key Concepts

Frequency Distribution

A frequency distribution organizes data into intervals or categories and shows how many data points fall into each. It helps summarize large data sets and is essential for calculating measures like the mean and standard deviation from grouped data.

Midpoint of Class Intervals

The midpoint is the value halfway between the lower and upper boundaries of a class interval. It represents the typical value for all data points in that interval and is used to approximate calculations such as the mean and standard deviation when only grouped data is available.

Standard Deviation for Grouped Data

Standard deviation measures the spread or variability of data around the mean. For grouped data, it is approximated by using class midpoints and frequencies to calculate the mean and then the squared deviations, weighted by frequency, to find the variance and standard deviation.