Problem 34 - Basic Statistics for Business and Economics

Question

A recent study of home technologies reported the number of hours of personal computer usage per week for a sample of 60 persons. Excluded from the study were people who worked out of their home and used the computer as a part of their work. $$ \begin{array}{|rrrrrrrrrr|} \hline 9.3 & 5.3 & 6.3 & 8.8 & 6.5 & 0.6 & 5.2 & 6.6 & 9.3 & 4.3 \\ 6.3 & 2.1 & 2.7 & 0.4 & 3.7 & 3.3 & 1.1 & 2.7 & 6.7 & 6.5 \\ 4.3 & 9.7 & 7.7 & 5.2 & 1.7 & 8.5 & 4.2 & 5.5 & 5.1 & 5.6 \\ 5.4 & 4.8 & 2.1 & 10.1 & 1.3 & 5.6 & 2.4 & 2.4 & 4.7 & 1.7 \\ 2.0 & 6.7 & 1.1 & 6.7 & 2.2 & 2.6 & 9.8 & 6.4 & 4.9 & 5.2 \\ 4.5 & 9.3 & 7.9 & 4.6 & 4.3 & 4.5 & 9.2 & 8.5 & 6.0 & 8.1 \\ \hline \end{array} $$ a. Organize the data into a frequency distribution. How many classes would you suggest? What value would you suggest for a class interval? b. Draw a histogram. Interpret your result.

Step-by-Step Solution

Answer

Organize data into 8 classes with intervals of 1.5 units. Histogram suggests most usage between 4.9 to 7.9 hours per week.

Step 1: Determine the Range and Number of Classes

To find the range, subtract the smallest value from the largest: \[ \text{Max} = 10.1, \text{Min} = 0.4, \text{Range} = 10.1 - 0.4 = 9.7 \]A common rule for the number of classes (k) is the Sturges' formula: \[k = 1 + 3.322 \log(n)\]where \(n = 60\). Calculate: \[k \approx 1 + 3.322 \log(60) \approx 1 + 3.322 \times 1.778 = 7.91 \approx 8\]

Step 2: Calculate Class Width

The class width \(w\) can be calculated by dividing the range by the number of classes and rounding up:\[w = \frac{\text{Range}}{\text{Number of classes}} = \frac{9.7}{8} \approx 1.2125 \]Round up to 1.5 to ensure all data is covered.

Step 3: Create Class Intervals

Start from the minimum value and add the class width to define intervals: - 0.4 to 1.9 - 1.9 to 3.4 - 3.4 to 4.9 - 4.9 to 6.4 - 6.4 to 7.9 - 7.9 to 9.4 - 9.4 to 10.9

Step 4: Count Frequencies for Each Class

Count the number of observations in each class: - 0.4 to 1.9: 8 - 1.9 to 3.4: 11 - 3.4 to 4.9: 12 - 4.9 to 6.4: 13 - 6.4 to 7.9: 10 - 7.9 to 9.4: 5 - 9.4 to 10.9: 1

Step 5: Draw the Histogram

Use the intervals and frequencies calculated to draw a histogram, with intervals on the x-axis and frequencies on the y-axis. Each bar's height corresponds to the frequency of the interval.

Step 6: Interpret the Histogram

The histogram shows that most people use their personal computers between 4.9 to 7.9 hours per week, indicating a central tendency around this range. Fewer people use the computer for extremely low or high amounts of time weekly.

Key Concepts

Histogram

Understanding how to visualize data is essential in statistics. In this exercise, a histogram is our tool of choice. A histogram is a type of bar graph used to represent the frequency distribution of a set of continuous or discrete data. Unlike a basic bar chart, the bars are adjacent to each other, indicating the data continues across intervals without gaps.

To create a histogram for the given data on computer usage hours, you need to define class intervals based on the data range and the number of data points or samples. Each interval corresponds to a bar in the histogram, with the bar height representing how many data points fall within that range or interval.

This visual representation helps students quickly grasp at which intervals the data points are concentrated, along with spotting any trends or patterns within the data sample.

Data Visualization

Data visualization is a critical aspect of data handling, as it helps interpret complex datasets into understandable graphics or charts. A histogram, as used in the given exercise, serves this purpose well by turning raw numerical data into a visual format that is easier to comprehend.

By doing this, one can easily identify the distribution, frequency, and central tendencies of the data. It communicates dense information clearly, allowing for the recognition of different data patterns, such as skewness or the presence of outliers.

• It simplifies complex numerical data into an inviting visual format.
• Enables quick initial analysis to identify trends.
• Aids in comparing different datasets through visual inspection.

Overall, data visualization is pivotal in providing insights that are immediately interpretable without deep statistical analysis.

Class Intervals

Class intervals are a fundamental concept in creating histograms and organizing data sets. In statistics, these intervals divide the entire range of data into segments or "bins," each representing different ranges of values.

In the histogram for computer usage data, seven class intervals are defined from 0.4 to 10.9 hours per week, with each interval width calculated to be 1.5. Choosing appropriate class intervals is crucial:

• They must cover the entire range of data without gaps or overlaps.
• Each interval width should be equal or as uniform as possible for consistency.
• The number of classes or intervals is generally determined using Sturges' formula or similar rules to strike a balance between detail and simplicity.

Well-defined class intervals lead to clear and informative histograms, ensuring accurate data interpretation.

Central Tendency

Central tendency refers to the measure(s) that represent a typical value in a dataset, indicating the center point of distribution. In our histogram example, the central tendency can be visually determined by observing where the data is most concentrated.

The histogram reveals that most subjects in the study use their personal computers between 4.9 and 7.9 hours per week. This marks the central tendency of the dataset, suggesting a pattern of typical usage behavior.

Central tendency can be quantified using:

• Mean: The average value of the dataset.
• Median: The middle value when data are ordered.
• Mode: The most frequently occurring value.

Understanding the central tendency provides valuable insights into general behavior or trends within the dataset and helps compare across different samples.