Question

The tables that follow report the marital status of 20 respondents in two different apartment complexes. (HINT: Make sure that you have the correct numbers in the numerator and denominator before answering these questions. For example, problem 2.1a asks for "the percentage of respondents in each complex who are married," and the denominators will be 20 for these two fractions. Problem \(2.1 d\), on the other band, asks for the "percentage of the single respondents who live in Complex \(B\)," and the denominator for this fraction will be \(4+6\), or \(10 .)\) \begin{tabular}{lcc} \hline \multicolumn{1}{c} { Status } & Complex A & Complex B \\ \hline Married & 5 & 10 \\ Unmarried & 8 & 2 \\ ("living together") & & \\ Single & 4 & 6 \\ Separated & 2 & 1 \\ Widowed & 0 & 1 \\ Divorced & \(\frac{1}{20}\) & \(\frac{0}{20}\) \end{tabular} a. What percentage of the respondents in each complex are married? b. What is the ratio of single to married respondents at each complex? c. What proportion of each sample are widowed? d. What percentage of the single respondents live in Complex B? e. What is the ratio of the "unmarried/living together" to the "married" at each complex?

Short answer

a) 25%, 50%; b) 0.8, 0.6; c) 0, 0.05; d) 60%; e) 1.6, 0.2.

Step-by-step solution

Percentage of Married Respondents

Calculate the percentage of married respondents in each complex using the formula: \[ \text{Percentage} = \left( \frac{\text{Number of Married Respondents}}{\text{Total Respondents}} \right) \times 100 \]For Complex A: \( \frac{5}{20} \times 100 = 25\% \)For Complex B: \( \frac{10}{20} \times 100 = 50\% \)

Ratio of Single to Married Respondents

Calculate the ratio of single to married respondents in each complex:For Complex A: Ratio = \( \frac{4}{5} = 0.8 \)For Complex B: Ratio = \( \frac{6}{10} = 0.6 \)

Proportion of Widowed Respondents

Calculate the proportion of widowed respondents in each complex using the formula:\[ \text{Proportion} = \left( \frac{\text{Number of Widowed Respondents}}{\text{Total Respondents}} \right) \]For Complex A: Proportion = \( \frac{0}{20} = 0 \)For Complex B: Proportion = \( \frac{1}{20} = 0.05 \)

Percentage of Single Respondents in Complex B

Calculate the percentage of single respondents who live in Complex B using the formula:\[ \text{Percentage} = \left( \frac{\text{Number of Single Respondents in Complex B}}{\text{Total Single Respondents}} \right) \times 100 \]Total Single Respondents = 4 + 6 = 10\( \text{Percentage} = \frac{6}{10} \times 100 = 60\% \)

Ratio of Unmarried/Living Together to Married

Calculate the ratio of unmarried/living together to married respondents in each complex:For Complex A: Ratio = \( \frac{8}{5} = 1.6 \)For Complex B: Ratio = \( \frac{2}{10} = 0.2 \)

Key concepts

Percentage Calculation

Percentage calculation is a fundamental and common procedure in statistics. To calculate the percentage of respondents who are married in each complex, we use the formula:
\[ \text{Percentage} = \left( \frac{\text{Number of Married Respondents}}{\text{Total Respondents}} \right) \times 100 \]
In Complex A, there are 5 married respondents out of 20 total respondents. So, the calculation for Complex A is: \[ \frac{5}{20} \times 100 = 25\% \]
In Complex B, there are 10 married respondents out of 20 total respondents. So, the calculation for Complex B is: \[ \frac{10}{20} \times 100 = 50\% \]
Knowing how to calculate percentages is essential because it helps place a value or portion relative to the whole. This is useful in understanding distributions, trends, and comparisons clearly.

Ratio Calculation

Ratios compare two quantities to understand their relative size to each other. To find the ratio of single to married respondents, we use the formula:
\[ \text{Ratio} = \frac{\text{Number of Single Respondents}}{\text{Number of Married Respondents}} \]
For Complex A, there are 4 single respondents and 5 married respondents. The ratio is: \[ \frac{4}{5} = 0.8 \]
For Complex B, there are 6 single respondents and 10 married respondents. The ratio is: \[ \frac{6}{10} = 0.6 \]
Ratios are crucial for portraying how much of one thing there is compared to another. They allow us to analyze relationships and proportions within datasets.

Proportion Calculation

Proportion is a type of ratio that expresses how a part relates to the whole. To calculate the proportion of respondents who are widowed, we use the formula:
\[ \text{Proportion} = \left( \frac{\text{Number of Widowed Respondents}}{\text{Total Respondents}} \right) \]
In Complex A, there are no respondents who are widowed. So, the calculation is: \[ \frac{0}{20} = 0 \]
In Complex B, there is 1 widowed respondent out of 20 total respondents. So, the calculation is: \[ \frac{1}{20} = 0.05 \]
Proportion is essential in statistics as it helps quantify the part of the dataset that fulfills a certain condition compared to the total. This is particularly useful in understanding demographic distributions.

Statistical Analysis

Statistical analysis involves the process of collecting and interpreting data to uncover patterns and trends. In this exercise, we calculated percentages, ratios, and proportions to draw meaningful insights. For example, observing the percentage of married respondents in each complex helps understand the demographic composition in these areas. Additionally, calculating the ratio of single to married respondents gives insight into the relative frequency of these statuses. Lastly, the proportion of widowed respondents quantifies a small, specific demographic. These analyses help in making informed decisions based on the data trends.

Sample Data Interpretation

Sample data interpretation is the process of making sense of numerical data obtained from a sample of a population. In this exercise, we used different statistical techniques to interpret marital status data from two apartment complexes. Calculating the percentage, ratio, and proportion helped us understand the specific characteristics and differences between Complex A and Complex B. For instance, we found that a higher percentage of respondents in Complex B are married compared to Complex A. We also used ratios to see how the number of single respondents compared to married respondents. Such interpretations are valuable in drawing conclusions from sample data and understanding broader trends within the population.