Constructing and Graphing Discrete Probability Distributions In Exercises 19 and 20, (a) construct a probability distribution, and (b) graph the probability distribution using a histogram and describe its shape.


Televisions The number of high-definition (HD) televisions per household in a small town

Verified step by step guidance

1. Step 1: Calculate the total number of households by summing the values in the 'Households' row. This will be used to determine the probabilities for the probability distribution.
2. Step 2: For each category of televisions (0, 1, 2, 3 or more), calculate the probability by dividing the number of households in that category by the total number of households. Use the formula: P(X = x) = (Number of households for x televisions) / (Total number of households).
3. Step 3: Construct the probability distribution table. The table should include two columns: one for the number of televisions (0, 1, 2, 3 or more) and one for the corresponding probabilities calculated in Step 2.
4. Step 4: Create a histogram to graph the probability distribution. On the x-axis, plot the number of televisions (0, 1, 2, 3 or more), and on the y-axis, plot the probabilities. Ensure the bars are proportional to the probabilities.
5. Step 5: Analyze the shape of the histogram. Determine whether the distribution is symmetric, skewed left, skewed right, or uniform based on the heights of the bars.

Key Concepts

Discrete Probability Distribution

A discrete probability distribution describes the likelihood of each possible outcome in a discrete sample space. In this context, it represents the number of high-definition televisions per household, where each outcome (0, 1, 2, or 3 or more televisions) has a corresponding probability calculated by dividing the number of households with that outcome by the total number of households.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. In this case, the histogram will display the frequency of households for each category of televisions, allowing for a visual interpretation of the probability distribution and helping to identify patterns or trends in the data.

Shape of the Distribution

The shape of a probability distribution provides insights into the data's characteristics, such as symmetry, skewness, and modality. For the given data on televisions per household, analyzing the histogram's shape will help determine whether the distribution is uniform, normal, skewed, or has multiple peaks, which can inform conclusions about household television ownership in the small town.