Problem 25 - Basic Statistics for Business and Economics

Question

The following is the number of minutes to commute from home to work for a sample of aerospace workers in Houston, Texas. $$ \begin{array}{|lllllllllllll|} \hline 28 & 25 & 48 & 37 & 41 & 19 & 32 & 26 & 16 & 23 & 23 & 29 & 36 \\ 31 & 26 & 21 & 32 & 25 & 31 & 43 & 35 & 42 & 38 & 33 & 28 & \\ \hline \end{array} $$ a. How many classes would you recommend? b. What class interval would you suggest? c. What would you recommend as the lower limit of the first class? d. Organize the data into a frequency distribution. e. Comment on the shape of the frequency distribution.

Step-by-Step Solution

Answer

a) 6 classes. b) Class interval of 6. c) Lower limit of 15. d) Frequency distribution: [15-20): 2, [20-26): 4, [26-32): 6, [32-38): 6, [38-44): 4, [44-50): 1. e) The distribution is fairly symmetrical.

Step 1 - Determine the Number of Classes

The number of classes for a frequency distribution can often be estimated using Sturges' formula: \( k = 1 + 3.322 \log_{10} n \), where \( n \) is the number of observations. In this case, count the number of data points: 25. Substitute into Sturges' formula: \[ k = 1 + 3.322 \log_{10} 25 \approx 1 + 3.322 \times 1.398 = 5.64 \]. Therefore, approximately 6 classes are recommended.

Step 2 - Determine the Class Interval

The class interval is determined by dividing the range by the number of classes. First, find the range: subtract the minimum value (16) from the maximum value (48). So, range = 48 - 16 = 32. Divide the range by the number of classes \( k = 6 \): \( \frac{32}{6} \approx 5.33 \). Round this to a convenient number, such as 6. So, the suggested class interval is 6.

Step 3 - Determine the Lower Limit of the First Class

Choose a lower limit that is slightly less than or equal to the smallest data value. The smallest value is 16, so a convenient lower limit for the first class could be 15.

Step 4 - Construct the Frequency Distribution

Using the lower limit of 15 and a class interval of 6, construct the classes: [15-20), [20-26), [26-32), [32-38), [38-44), and [44-50). Count how many data points fall into each class to get the frequency for each interval: - [15-20): 2 - [20-26): 4 - [26-32): 6 - [32-38): 6 - [38-44): 4 - [44-50): 1.

Step 5 - Describe the Shape of the Distribution

After examining the frequency distribution, it appears to be fairly symmetrical. The distribution does not skew heavily to the left or right, meaning there is a balanced spread of data around the central classes.

Key Concepts

Sturges' formula

Sturges' formula is a helpful tool in statistics used to determine the number of classes or intervals for a frequency distribution. This formula is important because it helps in organizing a large dataset into a more digestible form. The formula is given by: \[ k = 1 + 3.322 \log_{10} n \]where \( k \) is the number of classes and \( n \) is the number of observations in your data set. Using Sturges' formula ensures that the data is neither over-clustered into too few classes nor overly diffused into too many which can obscure patterns within the data.For the aerospace workers' commute time dataset, with 25 data points, the formula helps us estimate approximately 6 classes (rounded from 5.64) for an effective frequency distribution.

class interval

The class interval is a crucial element when creating a frequency distribution, as it determines the width of each class. To find an appropriate class interval, you need to calculate the range of your data. This is done by subtracting the smallest value from the largest value. For example, in the given dataset, the range is calculated as:\[ \text{Range} = 48 - 16 = 32 \]Once you have the range, divide it by the number of classes derived from Sturges' formula, in this case, approximately 6. Hence, the calculation is:\[ \text{Class Interval} = \frac{32}{6} \approx 5.33 \]Rounding up to a more convenient number, such as 6, allows for neatly defined classes that effectively encompass all data points within the range.

histogram

A histogram is a graphical representation of a frequency distribution with bars. It requires the information from class intervals and frequencies to construct. The height of each bar represents the frequency of data points within each class. In constructing the histogram from the aerospace workers' commute times, bars are drawn for each class like - [15-20) - [20-26) - ... up to [44-50) The height corresponds to how many data points fall into each interval:

• [15-20): 2
• [20-26): 4
• ...
• [44-50): 1

A well-organized histogram can reveal the shape of the distribution, indicating where most data points are clustered or if there is any trend. It is a powerful visual tool in data analysis to easily identify patterns or anomalies in datasets.

data analysis

Data analysis involves examining, transforming, and arranging data to present it in an informative way, provide insights, and guide decision-making. In this exercise, data analysis begins with calculating the number of classes and the class interval using Sturges' formula, then proceeds by organizing data into a frequency distribution. This overview not only simplifies the dataset's complexity but also uncovers patterns that are not immediately obvious. Visual tools like histograms, produced from this frequency distribution, further aid data interpretation by illustrating data shapes, showing symmetry, skewness, and central tendencies. Converting numbers into a structured layout extracts meaning from originally raw data, making data more comprehensible and useful in practical scenarios. Through these steps, data analysis provides a strategic glimpse into how aerospace workers distribute their commuting times, offering both granular and holistic insights.