Question

A survey asks 60 teachers and 48 parents whether school uniforms reduce distractions in school. Of those, 49 teachers and 18 parents say uniforms reduce distractions in school. Organize these results in a two-way table. Then find and interpret the marginal frequencies.

Short answer

The two-way table representation shows that out of 60 teachers, 49 believe school uniforms reduce distractions while 11 do not. Out of 48 parents, 18 believe uniforms reduce distractions while 30 do not. This means that a major percentage of teachers (81.7%) compared to parents (37.5%) believe that uniforms reduce distractions. In total, approximately 62% of all respondents believe that uniforms reduce distractions.

Step-by-step solution

Creation of a Two-Way Table

First, create a two-way table that represents the collected data. The rows will represent the two groups in the survey, 'Teachers' and 'Parents', while the columns will represent the two possible responses, 'Believe Uniforms Reduce Distractions' and 'Don't Believe Uniforms Reduce Distractions'. The cell at the intersection of each row and column will contain the number of individuals from each category that gave a specific response. In this case, based on the given information, fill in the cells as follows: \n\n| | Believe | Don't Believe | Total | \n| :--- | ---: | ---: | ---: | \n| Teachers | 49 | 11 | 60 | \n| Parents | 18 | 30 | 48 | \n| Column Total| 67 | 41 | 108 |

Calculation of Marginal Frequencies

Next, calculate the marginal frequencies. In this two-way table, the marginal frequencies are the totals represented in the last row and column. They represent the total for each group and the total for each response. These are already calculated in the table in Step 1: \n\nTeachers who believe uniforms reduce distractions: 49\nTeachers who don't believe uniforms reduce distractions: 11 \nParents who believe uniforms reduce distractions: 18 \nParents who don't believe uniforms reduce distractions: 30 \nTotal respondents who believe uniforms reduce distractions: 67 \nTotal respondents who don't believe uniforms reduce distractions: 41

Interpretation of Marginal Frequencies

The marginal frequencies provide useful information about the overall opinions on school uniforms from the surveyed parents and teachers. From the above frequencies, it can be interpreted that a larger percentage of teachers (49 out of 60 total teachers, or approximately 81.7%) believe that uniforms reduce distractions compared to parents (18 out of 48 total parents, or approximately 37.5%). Additionally, out of 108 total respondents, 67 (or approximately 62%) believe that uniforms reduce distractions, while 41 (or approximately 38%) do not believe that uniforms reduce distractions.

Key concepts

Marginal Frequencies

Marginal frequencies in a two-way table are the totals found in the margins, which summarize the data of an entire row or column. In survey analysis, these numbers tell us how many total responses belong to a particular category. For example, in the two-way table from our exercise, these frequencies are located at the end of each row and column.

The marginal frequencies help us understand the general trends and overall distribution of responses. They provide a quick way to see how many respondents belong to each category. By summing up the rows and columns, one can discern the grand total of responses, affording a snapshot of the data set as a whole.

• Rows represent groups like Teachers and Parents.
• Columns represent response options like "Believe" or "Don't Believe".
• Marginal frequencies help highlight significant patterns in the data.

It's essential to accurately calculate these as they form the basis for deeper data analysis, allowing comparisons between different groups and their responses.

Data Organization

Data organization is crucial when working with survey data because it transforms raw information into a structured format that's easier to interpret. In our exercise, a two-way table was used to organize data. This type of table is particularly effective for categorizing data with two distinct categories or variables, such as participant groups and their responses in surveys.

With clear rows (representing participants: teachers and parents) and columns (responses: "Believe" and "Don't Believe"), the data becomes much easier to digest. Organizing data in a two-way table:

• Provides a clear visual snapshot of the results.
• Allows for easy calculation of totals and percentages.
• Facilitates straightforward comparison across different groups and responses.

Effective data organization reveals patterns and relationships in the survey data, making statistical interpretation more intuitive.

Survey Analysis

Survey analysis is a process that involves the evaluation and interpretation of survey results. In our example, the analysis centers around responses from two groups – teachers and parents – about school uniforms. By creating a two-way table, we were able to examine the proportions and trends within the data more clearly.

The key steps in survey analysis using a two-way table include:

• Identifying the groups and their responses.
• Calculating marginal frequencies to understand overall data distribution.
• Analyzing these frequencies to discover trends and patterns, such as the difference in perspectives between groups.

Survey analysis allows for the understanding of opinions and behaviors within the surveyed populations, providing important insights that lead to better decision-making or further studies.

Response Interpretation

Interpreting survey responses accurately is essential to understanding what the data truly conveys. For the given survey about school uniforms, response interpretation shapes our understanding of how strongly each group feels about the issue.

Consider the results when comparing teachers and parents:

• Approximately 81.7% of teachers believe uniforms reduce distractions.
• Only about 37.5% of parents share this belief.
• Overall, 62% of all respondents think uniforms are beneficial.

These interpretations provide valuable insights which can inform future decisions about school uniform policies. They reflect not just the numbers, but also the context and sentiment underlying the data, enabling stakeholders to act based on informed perspectives rather than assumptions.