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. $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ If a measurement is selected at random from the 20 measurements shown, what is the probability that it is a \(2 ?\)

Short answer

Answer: The probability of randomly selecting a measurement with a value of 2 is 0.25.

Step-by-step solution

Count the occurrences of each unique value

We will count the number of times each unique value (0, 1, and 2) appears in the 20 measurements. We can do this by looking at the columns and rows and tallying the occurrences of each value. 0: 6 appearances 1: 9 appearances 2: 5 appearances

Calculate the relative frequencies

To find the relative frequencies, we will divide the number of appearances of each unique value by the total number of measurements (20). This will give us the relative frequencies for each value 0, 1, and 2. Relative Frequency of 0: \(\frac{6}{20} = 0.30\) Relative Frequency of 1: \(\frac{9}{20} = 0.45\) Relative Frequency of 2: \(\frac{5}{20} = 0.25\)

Construct the relative frequency histogram

Using the relative frequencies calculated, we can construct a relative frequency histogram. The histogram will have three bars corresponding to the values 0, 1, and 2, and their height will represent the relative frequencies (0.30, 0.45, and 0.25 respectively). 1. Draw a horizontal axis with the discrete values (0, 1, and 2). 2. Draw vertical bars above each value on the horizontal axis with heights corresponding to the relative frequencies: - Value 0: height = 0.30 - Value 1: height = 0.45 - Value 2: height = 0.25

Calculate the probability of randomly selecting a measurement with a value of 2

To find the probability of selecting a random measurement with a value of 2, we can use the relative frequency of 2, as the relative frequency represents the proportion of the data set with that specific value. The relative frequency of 2 is \(0.25\), which means that the probability of randomly selecting a measurement with a value of 2 is \(\boxed{0.25}\).

Key concepts

Discrete Variable

In the realm of statistics and probability, a discrete variable is a type of variable that can take on a countable number of distinct values. Unlike continuous variables, which can assume any value within a range, discrete variables have specific, separate values. Examples of discrete variables include the number of cars in a parking lot, the number of students in a class, and, as in our exercise, the output values of a measurement that can only be 0, 1, or 2.

Understanding discrete variables is crucial when analyzing data or constructing graphs such as histograms because the nature of the variable influences both the type of data analysis that is appropriate and the way data is visually represented. In educational exercises, discrete variables often underpin explorations into probability, helping students to discern the difference between possible outcomes that are countable (discrete) versus those that are not (continuous).

Probability Distribution

The concept of a probability distribution is foundational in the understanding of statistical analysis for discrete variables. It lists all the possible outcomes of a random variable and the probabilities that these outcomes will occur. These probabilities must sum to 1 since one of the outcomes will definitely occur.

Consider a basic example: the roll of a fair six-sided die. The probability of rolling any given number between 1 and 6 is exactly 1/6 since the die has six faces, and we assume it is not biased. Therefore, the probability distribution of the die roll is a discrete uniform distribution, where each outcome has an equal chance of occurring. In the exercise we're considering, the probability distribution was determined for a variable with possible values of 0, 1, and 2. Analyzing probability distributions helps students not only in calculating the likelihood of certain events but also in understanding the concepts of expectation, variance, and standard deviation.

Histogram Construction

A histogram is a common graphical representation of the distribution of numerical data, and it is especially useful in depicting the frequency of data intervals for continuous data. However, when these histograms represent discrete data, they are transformed into relative frequency histograms.

The steps to build a relative frequency histogram are straightforward but must be carefully followed:

• Identify the range of data and the discrete variables.
• Calculate the relative frequency for each discrete variable by dividing the count of that variable by the total count of all data points.
• Construct a bar chart where each discrete variable is represented on the horizontal axis, and the height of the bar corresponds to the relative frequency.

Drawing a histogram aids in visualizing the data distribution and makes it easier for students to comprehend clustering of data, central tendency, and dispersion. Creating a relative frequency histogram from the dataset in our problem helped to visually represent the relative frequency of selecting different measures, thereby enhancing understanding of how likely each measurement was to occur.