Question

A discrete variable can take on only the values \(0,1,\) or 2 . A set of 20 measurements on this variable is shown here: $$\begin{array}{lllll}1 & 2 & 1 & 0 & 2 \\\2 & 1 & 1 & 0 & 0 \\\2 & 2 & 1 & 1 & 0 \\\0 & 1 & 2 & 1 & 1\end{array}$$ a. Construct a relative frequency histogram for the data. b. What proportion of the measurements are greater than \(1 ?\) c. What proportion of the measurements are less than \(2 ?\) d. If a measurement is selected at random from the 20 measurements shown, what is the probability that it is a \(2 ?\) e. Describe the shape of the distribution. Do you see any outliers?

Short answer

Question: Using the steps provided, describe the process of analyzing a set of 20 discrete measurements with values of 0, 1, and 2. Then, explain how to find the proportion of measurements greater than 1, less than 2, and the probability of selecting a 2. Additionally, describe the shape of the distribution and explain how to determine the presence of outliers. Answer: To analyze the dataset, first count the frequency of each value (0, 1, and 2) in the set of 20 measurements. Then, calculate the relative frequency for each value by dividing its frequency by the total number of measurements. Next, construct a histogram with values on the x-axis and relative frequencies on the y-axis. To find the proportion of measurements greater than 1, divide the frequency of value 2 by the total number of measurements. Similarly, find the proportion of measurements less than 2 by adding the frequencies of values 0 and 1, and dividing the sum by the total number of measurements. The probability of selecting a 2 at random is equal to its relative frequency. After constructing the histogram, observe its general shape (symmetrical, skewed left, or skewed right). In this case, there's no possibility of outliers since the dataset only includes three possible values.

Step-by-step solution

Counting the data points

Go through the list of 20 measurements and count the number of occurrences for each value (0, 1, and 2). Step 2: Calculating relative frequency

Relative frequencies

Divide the count for each value (from Step 1) by the total number of measurements (20) to obtain the relative frequency for each value. Step 3: Constructing a relative frequency histogram

Histogram construction

Create a histogram with the x-axis representing the discreet values (0, 1, and 2) and the y-axis representing the relative frequencies (found in Step 2). The bars in the histogram should be of equal width. Step 4: Proportion of measurements greater than 1

Proportion > 1

Using the frequency of 2 from Step 1, divide the frequency of 2 by the total number of measurements (20) to find the proportion of measurements greater than 1. Step 5: Proportion of measurements less than 2

Proportion < 2

Using the frequencies of 0 and 1 from Step 1, add their frequencies together and divide the sum by the total number of measurements (20) to find the proportion of measurements less than 2. Step 6: Probability of selecting a 2 at random

Probability of selecting a 2

The probability of selecting a 2 is equal to its relative frequency. Use the relative frequency of 2 from Step 2 as the probability of selecting a 2 at random. Step 7: Describing the shape of the distribution and identifying outliers

Shape and outliers

Observe the shape of the histogram from Step 3 and describe its general shape (symmetrical, skewed left, or skewed right). Since the dataset only includes three possible values, there's no possibility of outliers in this case.

Key concepts

Relative Frequency Histogram

A relative frequency histogram is a type of graph that displays the relative frequencies of different discrete values in a dataset. It's similar to a regular histogram but uses the percentages (or proportions) of the total number of data points instead of the actual counts. To create a relative frequency histogram, one must first count the occurrences of each discrete value. In our exercise, the values are 0, 1, and 2.
After counting, each frequency is divided by the total number of measurements to convert the counts into relative frequencies. For instance, if the value 1 appears 10 times in a 20-measurement set, its relative frequency would be \( \frac{10}{20} = 0.5 \), or 50%. The final step involves plotting these relative frequencies on the y-axis against the discrete values on the x-axis.

• The bars represent the discrete values.
• The height of each bar corresponds to the relative frequency of each value.

This visual representation is crucial for quickly identifying trends and patterns in the data distribution, such as which values occur most or least frequently.

Probability Distribution

Probability distribution is at the heart of understanding discrete variables in statistics. It represents the likelihood of each possible outcome in an experiment or a process. For discrete variables, a probability distribution assigns a probability to each possible value that the variable can take on.
In our textbook example, the discrete variable can take on the values 0, 1, or 2. Therefore, the probability distribution will specify the probability of each of these values occurring. To determine these probabilities, we use the relative frequency from our dataset as an estimator. The probability of selecting a particular value, say 2, randomly from the data is equal to the relative frequency of that value. If 2 appears 6 times out of 20 measurements, the probability of randomly selecting a 2 is \( \frac{6}{20} = 0.3 \), or 30%.
Recognizing the probability distribution allows us to predict outcomes and quantify the randomness of events in quantitative studies. It's a conceptual bedrock for inferential statistics and related areas, aiding in decision making and hypothesis testing.

Data Analysis

Data analysis involves inspecting, cleansing, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. It encompasses a variety of techniques and processes applied to obtain insights from data. In the context of our exercise, data analysis includes organizing raw data (the set of 20 measurements), computing relative frequencies, and interpreting these frequencies to create a probability distribution.
Analyzing the proportions for measurements greater than 1 or less than 2, we can draw conclusions about the data set. For instance, if we find that the measurements greater than 1 constitute a high proportion, this may indicate a tendency toward higher values in this particular data set. Through data analysis:

• We gain a better understanding of the underlying structure of the data.
• We can make predictions based on the data.
• We identify the shape of the distribution (such as symmetrical or skewed) to infer about the central tendency and variability.

These steps and interpretations help transform raw data into actionable insights, a crucial aspect in fields ranging from business to healthcare.