Question

Construct a relative frequency histogram for these 20 measurements on a discrete variable that can take only the values \(0, I,\) and 2. Then answer the questions. $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ What proportion of the measurements are greater than \(1 ?\)

Short answer

Answer: The proportion of measurements greater than 1 in the given data set is 0.25 or 25%.

Step-by-step solution

Organize the data

First, let's organize the given data into a frequency table. This will help us visualize the data frequency better. | Value | Frequency | |-------|-----------| | 0 | 6 | | 1 | 9 | | 2 | 5 |

Compute the relative frequencies

Now, we will calculate the relative frequency of each value by dividing its frequency by the total number of measurements (20). | Value | Frequency | Relative Frequency | |-------|-----------|--------------------| | 0 | 6 | 6/20 = 0.3 | | 1 | 9 | 9/20 = 0.45 | | 2 | 5 | 5/20 = 0.25 |

Construct the histogram

Construct a histogram with the discrete values (0, 1, and 2) on the x-axis and the relative frequencies on the y-axis. The relative frequency for each value will be represented by the height of the corresponding bar. |-------| |2: |_ | |____ |1: |_ | |____ |0: |_| |-------|

Determine proportion greater than 1

To find the proportion of measurements greater than 1, we will sum up the relative frequencies of the values greater than 1 (in our case, it's only the value 2). Proportion of measurements > 1 = Relative Frequency of 2 = 0.25 Hence, the proportion of measurements greater than 1 is 0.25 (25%).

Key concepts

Discrete Variable

A discrete variable is a type of variable that can take on a finite or countable number of values. The values that a discrete variable can take are distinct and separate; there are no in-between values. For instance, when considering the outcome of rolling a six-sided die, the die can end up showing one of six numbers: 1, 2, 3, 4, 5, or 6. There's no chance of rolling a 3.5 or any other value between the integers.

In the exercise, the discrete variable can only take the values 0, 1, and 2. These are akin to categories each measurement can fall into, making it easy to count the number of occurrences of each category. This countability makes discrete variables particularly suitable for statistical analysis using frequency tables and histograms, as shown in the exercise.

Frequency Table

A frequency table is a tool that allows you to organize and display data so it can be easily analyzed. It shows how often each value in a set of data occurs; in other words, it tabulates the frequency of different values. The frequency table for the given exercise, for example, listed how many times each of the discrete values (0, 1, and 2) appeared in the dataset.

To create a frequency table, you simply tally the occurrences of each discrete value. In the exercise, the frequency table was a stepping stone to find relative frequencies, which provide insight into the proportion of each category relative to the whole dataset.

Histogram Construction

Converting data from a frequency table into a histogram visualizes the distribution of data across different categories. When constructing a histogram, each category (or value) is represented on the x-axis, while the y-axis stands for the frequency or relative frequency of those values. Bars of different heights are used to represent how often each value occurs.

In the exercise, the histogram was constructed by plotting the discrete values 0, 1, and 2 on the x-axis with their corresponding relative frequencies on the y-axis. The height of each bar correlates with how often that value occurred relative to the total number of measurements. Visual aids like histograms make it easier to comprehend the underlying distribution and frequency of data points within a set.

Probability and Statistics

Probability and statistics are branches of mathematics dealing with data analysis. Probability provides a measure of how likely it is for a certain event to occur, while statistics is the practice of collecting, analyzing, interpreting, presenting, and organizing data.

In the context of the exercise, statistics were used to analyze a dataset and present this analysis in the form of a frequency table and a histogram. Probability concepts come into play when interpreting results, such as determining the proportion of measurements that are greater than 1. This outcome (25%) was interpreted as the probability that a randomly selected measurement from the dataset would be greater than 1.