Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A survey of 1000 families was conducted by the Public Housing Authority in a certain community to determine the distribution of families by size. The results follow: $$\begin{array}{lccccccc}\hline \text { Family Size } & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text { Frequency of } & & & & & & & \\ text { Occurrence } & 350 & 200 & 245 & 125 & 66 & 10 & 4 \\ \hline\end{array}$$ a. Find the probability distribution of the random variable \(X\), where \(X\) denotes the number of persons in a randomly chosen family. b. Draw the histogram corresponding to the probability distribution found in part (a).

Step-by-Step Solution

Answer

The probability distribution for the random variable X is: \(P(X=k) = \begin{cases} 0.35, & \text{if } k=2 \\ 0.2, & \text{if } k=3 \\ 0.245, & \text{if } k=4 \\ 0.125, & \text{if } k=5 \\ 0.066, & \text{if } k=6 \\ 0.01, & \text{if } k=7 \\ 0.004, & \text{if } k=8 \\ \end{cases}\) To draw the histogram, use the probabilities calculated for each family size. The x-axis represents the family size and the y-axis represents the probability. Plot bars of different heights according to the probabilities and make sure the bars are of even width and are adjacent to each other.

Step 1: Find the Total Number of Families in the Survey

There are a total of 1000 families surveyed, which will serve as our denominator for probability calculations.

Step 2: Find the Normalized Frequency for Each Family Size

To find the normalized frequency, or probability, for each family size, we divide the frequency of occurrence by the total number of families. For family size 2: \(P(X=2) = \frac{350}{1000} = 0.35\) For family size 3: \(P(X=3) = \frac{200}{1000} = 0.2\) For family size 4: \(P(X=4) = \frac{245}{1000} = 0.245\) For family size 5: \(P(X=5) = \frac{125}{1000} = 0.125\) For family size 6: \(P(X=6) = \frac{66}{1000} = 0.066\) For family size 7: \(P(X=7) = \frac{10}{1000} = 0.01\) For family size 8: \(P(X=8) = \frac{4}{1000} = 0.004\)

Step 3: Write Down the Probability Distribution

We can now write down the probability distribution for the random variable X: \(P(X=k) = \begin{cases} 0.35, & \text{if } k=2 \\ 0.2, & \text{if } k=3 \\ 0.245, & \text{if } k=4 \\ 0.125, & \text{if } k=5 \\ 0.066, & \text{if } k=6 \\ 0.01, & \text{if } k=7 \\ 0.004, & \text{if } k=8 \\ \end{cases}\)

Step 4: Draw the Histogram

To draw the histogram, we use the probabilities calculated in step 2 for each family size. The x-axis represents the family size and the y-axis represents the probability. Use bars of different heights according to the probabilities calculated. For example, a bar representing family size 2 would have a height of 0.35, a bar representing family size 3 would have a height of 0.2, and so on. Make sure the bars are of even width and are adjacent to each other. Now, with the above histogram plot, we can visualize the probability distribution of the number of persons in a randomly chosen family.

Key Concepts

Random Variable

In statistics and probability, a random variable is a numerical value determined by the outcome of a random phenomenon. In this exercise, the random variable, denoted as \(X\), represents the number of persons in a family, randomly selected from a survey of 1000 families.

Each family size is a potential outcome for the random variable \(X\). These outcomes range from a family of 2 persons to a family of 8 persons. The exercise helps to understand how often each family size occurs, which is a key feature when analyzing random variables in real-world settings.

By calculating the probability distribution for this variable, we can see which family sizes are more common. This step involves dividing the frequency of each family size by the total number of families, providing insights into the likelihood of each outcome.

Histogram

A histogram is a type of bar graph representing the frequency distribution of numerical data. It helps us visualize how data is distributed.

In this exercise, the histogram is based on the probability distribution of family sizes. On the x-axis, you list the family sizes (from 2 to 8), and on the y-axis, you have the probabilities we calculated. Each bar's height corresponds to the probability of each family size.

For instance, the bar for a family size of 2 would reach 0.35 on the y-axis, while the bar for a family size of 3 would stretch to 0.2. The histogram provides a visual summary of which family sizes are most and least common, allowing for simple comparison of probabilities across the different family sizes.

Probability Calculation

Probability calculations help us determine how likely specific events are in a given context. In our exercise, we look at the probability of a family having a certain number of members based on survey data.

To calculate the probability \(P(X=k)\) where \(k\) is a given family size, you divide the frequency of families of size \(k\) by the total number of families surveyed. This calculation results in normalized frequencies or probabilities.

• For family size 2, the probability is \(\frac{350}{1000} = 0.35\).
• For family size 3, the probability is \(\frac{200}{1000} = 0.2\).
• For family size 4, it is \(\frac{245}{1000} = 0.245\).
• Continuing this process provides probabilities for each family size.

These calculations reveal not only the likelihood of selecting a family of any particular size but also show the overall distribution across family sizes.

Family Survey Analysis

Family survey analysis allows us to gather insights from data collected in surveys about households. The Public Housing Authority surveyed 1000 families to determine their sizes. The task was to analyze this data to find trends and patterns.

By organizing the data into a frequency distribution, we can calculate the probabilities for each family size and create a histogram for visual analysis. This kind of survey analysis helps identify areas where resources might be needed or adjustments in public policy under family-related circumstances.

For example, if a large number of small size families are observed, community resources like housing or social programs can be adjusted to meet their needs. The results from survey analysis drive data-informed decisions and facilitate better planning.