91Ó°ÊÓ

The Nielsen Home Technology Report provides information about home technology and its usage. The following data are the hours of personal computer usage during one week for a sample of 50 persons. \\[ \begin{array}{rrrrrrrr} 4.1 & 1.5 & 10.4 & 5.9 & 3.4 & 5.7 & 1.6 & 6.1 & 3.0 & 3.7 \\ 3.1 & 4.8 & 2.0 & 14.8 & 5.4 & 4.2 & 3.9 & 4.1 & 11.1 & 3.5 \\ 4.1 & 4.1 & 8.8 & 5.6 & 4.3 & 3.3 & 7.1 & 10.3 & 6.2 & 7.6 \\ 10.8 & 2.8 & 9.5 & 12.9 & 12.1 & 0.7 & 4.0 & 9.2 & 4.4 & 5.7 \\ 7.2 & 6.1 & 5.7 & 5.9 & 4.7 & 3.9 & 3.7 & 3.1 & 6.1 & 3.1 \end{array} \\] Summarize the data by constructing the following: a. A frequency distribution (use a class width of three hours) b. \(\quad\) A relative frequency distribution c. \(\quad\) A histogram d. An ogive e. Comment on what the data indicate about personal computer usage at home.

Step by step solution

01

Determine the Range and Number of Classes

First, we calculate the range of the data by subtracting the smallest value from the largest value. The smallest value is 0.7 and the largest is 14.8, giving a range of 14.1. Using a class width of 3 hours, we divide this range by the width to get the number of classes: \( \text{Number of classes} = \frac{14.1}{3} \approx 5 \) classes. We choose to use 6 classes to cover the entire range.

02

Construct the Frequency Distribution

Using the class width of 3, we can start at 0 and form the classes: [0-3), [3-6), [6-9), [9-12), [12-15). Count how many data values fall into each class and list them in a frequency distribution table: | Class | Frequency | |-------|-----------| | [0-3) | 8 | | [3-6) | 22 | | [6-9) | 10 | | [9-12) | 6 | | [12-15)| 4 |

03

Create the Relative Frequency Distribution

The total number of data points is 50. For each class, divide the class frequency by 50 to get the relative frequency: | Class | Relative Frequency | |-------|--------------------| | [0-3) | 0.16 | | [3-6) | 0.44 | | [6-9) | 0.20 | | [9-12) | 0.12 | | [12-15)| 0.08 |

04

Draw the Histogram

A histogram is a bar chart representing frequency distribution. On the x-axis, label the class intervals [0-3), [3-6), [6-9), [9-12), [12-15). The y-axis represents the frequency. Draw bars for each class to visualize the distribution.

05

Construct an Ogive

An ogive is a line graph of cumulative frequencies. First, calculate cumulative frequencies: | Upper Class Boundary | Cumulative Frequency | |----------------------|----------------------| | 3 | 8 | | 6 | 30 | | 9 | 40 | | 12 | 46 | | 15 | 50 | Plot these points on a graph, with the upper class boundaries on the x-axis and cumulative frequency on the y-axis. Connect the points with straight lines to complete the ogive.

06

Interpretation of Results

The distribution shows most participants used a computer between 3 and 6 hours. Very few were on the extreme ranges (0-3 and 12-15 hours). This indicates a moderate usage of computers at home, often short intervals rather than long continuous usage.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Histogram

A histogram is a powerful visual tool used to represent the frequency distribution of numerical data. Essentially, it is a bar graph that displays the number of observations, known as frequency, that fall within predefined intervals, called classes. In this exercise, we used a class width of 3 hours to organize the data into intervals: [0-3), [3-6), [6-9), [9-12), and [12-15).

Each bar corresponds to a class interval, with its height representing the frequency of data points in that class. In the histogram created from our data, we found that the highest bar corresponds to the interval [3-6), indicating that most people used their personal computers for between 3 to 6 hours during the week.

• X-axis: Represents the class intervals.
• Y-axis: Represents the frequency of data points in each class.
• Bars: Height is proportional to the class frequency.

This visual representation helps in quickly identifying where most data lie and spotting patterns or outliers in usage behavior.

Ogive

An ogive, also known as a cumulative frequency graph, provides insight into the number of occurrences below a particular value in a dataset. It is particularly beneficial for understanding the cumulative distribution of data, helping to see the percentage or number of items that fall below certain values.

To construct an ogive, we start by calculating cumulative frequencies. This involves summing the frequencies of all classes up to and including a given class. For the given data, we created the ogive using the upper boundaries of each class interval [0-3), [3-6), [6-9), [9-12), [12-15).

• X-axis: Displays the upper class boundaries (3, 6, 9, 12, 15).
• Y-axis: Shows cumulative frequency (8, 30, 40, 46, 50).
• Line: Connects plotted points, showing how frequency accumulates across classes.

This graph is informative for assessing percentile ranks and deducing median values, providing a straightforward way to analyze the spread and central tendencies of data.

Relative Frequency Distribution

The relative frequency distribution is a variation of the frequency distribution that shows the proportion of data points that fall into each class relative to the total number of observations. It is essentially each class's frequency divided by the total number of data points, which in our exercise is 50.

Showing the data in terms of relative frequency provides insight into the proportion of the dataset each class comprises, rather than just raw counts. For example, if the class [3-6) has a frequency of 22, the relative frequency is calculated as \( \frac{22}{50} = 0.44 \), indicating 44% of data points fell into this class.

• Provides a clearer picture of data distribution.
• Helps compare different datasets with varying totals.
• Useful when analyzing probabilities and statistical significance.

Relative frequencies can help quickly assess which class dominates the dataset, offering a more normalized view of the data.

Data Interpretation

Data interpretation is the process of making sense of numerical data that has been collected, analyzed, and presented. It is the step where you derive meaning from the graphs and computed statistics to make informed judgments or predictions.

For this exercise, interpreting the histogram and the ogive of personal computer usage data can lead us to conclude that a majority of the population uses their personal computers for a moderate amount of time, mainly around 3 to 6 hours per week. The relative absence of data points on the either extreme of the spectrum (i.e., very low usage or very high usage) suggests a balanced usage habit among the group studied.

• Assessing patterns: Most users fall in the moderate range.
• Identifying trends: Long continuous usage is rare.
• Predicting behavior: Gives an idea of typical usage behaviors.

Through thorough interpretation, insights can be drawn into user behavior trends, which are crucial for strategic decision-making in technology development and service provision.