91Ó°ÊÓ

The Nielsen Home Technology Report provided 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. \(\quad\) A frequency distribution (use a class width of three hours) b. A relative frequency distribution c. A histogram d. An ogive e. Comment on what the data indicate about personal computer usage at home.

Step by step solution

01

Organize the Data

Sort the data set in ascending order to prepare for the construction of a frequency distribution. The sorted data is as follows: 0.7, 1.5, 1.6, 2.0, 2.8, 3.0, 3.1, 3.1, 3.1, 3.3, 3.4, 3.5, 3.7, 3.7, 3.9, 3.9, 4.0, 4.1, 4.1, 4.1, 4.1, 4.2, 4.3, 4.4, 4.7, 4.8, 5.4, 5.6, 5.7, 5.7, 5.7, 5.9, 5.9, 6.1, 6.1, 6.1, 6.2, 7.1, 7.2, 7.6, 8.8, 9.2, 9.5, 10.3, 10.4, 10.8, 11.1, 12.1, 12.9, 14.8.

02

Create Frequency Distribution

Create a frequency distribution using a class width of 3 hours. Starting from the lowest value (0.7), classes would be: 1. 0 to 2.9 2. 3 to 5.9 3. 6 to 8.9 4. 9 to 11.9 5. 12 to 14.9 Count how many data points fall into each class: 1. 0 - 2.9: 5 2. 3 - 5.9: 22 3. 6 - 8.9: 7 4. 9 - 11.9: 9 5. 12 - 14.9: 7

03

Calculate Relative Frequencies

Calculate the relative frequency for each class by dividing the frequency by the total number of observations (50). 1. 0 - 2.9: \( \frac{5}{50} = 0.10 \)2. 3 - 5.9: \( \frac{22}{50} = 0.44 \)3. 6 - 8.9: \( \frac{7}{50} = 0.14 \)4. 9 - 11.9: \( \frac{9}{50} = 0.18 \)5. 12 - 14.9: \( \frac{7}{50} = 0.14 \)

04

Draw the Histogram

Draw a histogram with classes on the x-axis and frequencies on the y-axis. For each class, draw a bar that represents the frequency. Make sure each bar covers exactly one class width, ensuring no gaps between the bars.

05

Draw the Ogive

Construct an ogive by plotting the cumulative frequency for each class. Cumulative frequency is the sum of frequencies up to the current class. Cumulative frequencies are: 1. 0 - 2.9: 5 2. 3 - 5.9: 27 3. 6 - 8.9: 34 4. 9 - 11.9: 43 5. 12 - 14.9: 50 Plot these cumulative frequencies at the upper class boundary and connect the points to form the ogive.

06

Interpret the Data

By observing the frequency distribution and histogram, we can see that the majority of the sample spends between 3 and 5.9 hours per week using a personal computer at home. The ogive indicates that 50% of the users spend less than 6 hours, suggesting that personal computer usage at home is relatively moderate.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Frequency Distribution

A frequency distribution is a table that organizes data into various classes, or intervals, showing the number of occurrences in each. To create one, you first need to sort your data in ascending order and define class intervals, which here have a width of 3 hours. Start with the smallest number and add your class width to determine the next boundary. For the data provided, the classes range from 0 to 14.9 hours, covering five intervals. Each class must encompass all possible values, ensuring no overlap. By counting how many data points fall into each class, you can then fill out the frequency for each class. This distribution is essential, as it offers a simplified view of the dataset, helping to visualize and interpret patterns within the sample swiftly.

Histogram

A histogram is a graphical representation of a frequency distribution. It's made up of adjoining bars that represent the frequency of data within each interval. On the x-axis, you plot the class intervals, while the y-axis represents the frequency count. Each bar corresponds to a class, with the height of the bar indicating the number of occurrences, helping to visually convey how data is distributed across different intervals. Histograms effectively expose the shape, spread, and central tendency of the data. For instance, in this exercise, it illustrates that the most frequent computer usage falls between 3 to 5.9 hours. This visual representation can help quickly identify data clusters, highlighting where data points concentrate the most.

Relative Frequency

Relative frequency divides the frequency of each class by the total number of observations, converting absolute numbers into a proportion or percentage of the whole. This calculation provides a clearer picture of how each class contributes to the total dataset, allowing for easier comparison across different datasets or groups of varying sizes. For example, if the frequency of a class is 10 out of 50, the relative frequency is calculated as \( \frac{10}{50} = 0.20 \), meaning this interval represents 20% of the total data. Relative frequencies reveal how common or rare certain intervals are within the context of the entire data set, making it easier to analyze and interpret the data's overall trend.

Ogive

An ogive is a type of cumulative frequency graph. It is plotted on a graph with the cumulative frequencies on the y-axis and the upper class boundaries on the x-axis. By connecting points formed by cumulative frequencies, an ogive emerges. This curve is helpful for determining medians, quartiles, and understanding the cumulative data distribution. In practical analysis, an ogive can show how many observations fall below a particular value. For instance, you could use it to find that a certain percentage of users spend less than a given number of hours on their personal computers. The ogive from the exercise reflects that around 50% of individuals log fewer than 6 hours using their computers weekly at home, identifying a moderate level of engagement.