Question

In Exercises 7 and 8, use the two-way table to create a two-way table that shows the joint and marginal relative frequencies. $$ \begin{array}{|l|c|c|c|} \hline & \text { Left } & \text { Right } & \text { Total } \\ \hline \text { Female } & 11 & 104 & 115 \\ \hline \text { Male } & 24 & 92 & 116 \\ \hline \text { Total } & 35 & 196 & 231 \\ \hline \end{array} $$

Short answer

The two-way table showing the joint and marginal relative frequencies is as follows:\[\begin{array}{|c|c|c|c|}\hline & \text { Left } & \text { Right } & \text { Total } \\hline \text { Female } & 0.048 & 0.450 & 0.498 \ \hline \text { Male } & 0.104 & 0.398 & 0.502 \ \hline \text { Total } & 0.152 & 0.848 & 1 \ \hline\end{array} \]

Step-by-step solution

Calculate the Joint Relative Frequencies

In order to calculate the joint relative frequencies, divide each number by the total number of observations. The total number of observations is 231. Therefore (using the decimal representation with 3 decimal places), for example, the joint relative frequency of the first cell which represents the left-handed females is \( \frac{11}{231} = 0.048 \). Carry out these calculations for every cell.

Calculate the Marginal Relative Frequencies

To find the marginal relative frequencies, add up the joint relative frequencies in each row and column. For example, the marginal relative frequency for females is \(0.048 + 0.450 = 0.498\), while for left-handed people it is \(0.048 + 0.104 = 0.152\). Carry out these calculations for every row and column.

Construct the Two-Way Table

Now that we have all the joint and marginal relative frequencies, construct a new two-way table with these values. There should be a row for each category (female, male, total), a column for each category (left, right, total), and the relative frequencies filling the cells.

Key concepts

Joint Relative Frequencies

Joint relative frequencies are a way to understand the relationship between two categorical variables. In a two-way table, each cell represents the frequency of one specific combination of the two variables. To find the joint relative frequency, you divide the frequency of the cell by the total number of observations. For example, in the given exercise, we have 11 left-handed females out of a total of 231 people. To find the joint relative frequency of left-handed females, we calculate \[\text{Joint Relative Frequency} = \frac{11}{231} \approx 0.048\]This means that approximately 4.8% of the whole group consists of left-handed females. These calculations are crucial as they show how different categories intersect and help identify patterns or relationships between them. Joint relative frequencies let you see the interaction and distribution between two variables in the overall dataset. Always remember, joint relative frequencies should add up to 1 if you consider all cells in the table, highlighting the importance of accurate calculation.

Marginal Relative Frequencies

Marginal relative frequencies provide insights into the distribution of individual variables, without considering their intersection with another variable. They are calculated by summing the joint relative frequencies for each row or column. This results in a table with totals that represent each variable's share of the entire population.For instance, to calculate the marginal relative frequency for females, you sum the joint relative frequencies found in the row for females. This is done by adding the frequencies of left-handed and right-handed females:\[\text{Marginal Relative Frequency for Females} = 0.048 + 0.450 = 0.498\]Thus, females comprise roughly 49.8% of the total population in the study. Similarly, you can calculate the marginal relative frequency for left-handed people:\[\text{Marginal Relative Frequency for Left-handed} = 0.048 + 0.104 = 0.152\]Marginal relative frequencies are useful for observing the weight or proportion of each category within the context of the dataset, enabling a broader understanding without the interference of another variable.

Data Analysis

Data analysis in the context of two-way tables involves examining the relationships and distributions of categorical data. The first step is calculating joint and marginal relative frequencies to understand how different variables interact and individually contribute to the dataset. Once these frequencies are known, creating a two-way table enables easier visualization and interpretation of data. This new table reveals patterns, correlations, or anomalies that might not be immediately obvious through raw data alone. By organizing data appropriately, analysts can more effectively communicate findings and support decision-making or hypothesis testing. Beyond just calculating frequencies, good data analysis often requires:

• Checking for total sums to ensure calculations are accurate.
• Inspecting potential data entry errors or irregularities.
• Considering the context to determine the significance of the patterns observed.

Ultimately, data analysis in this form leads to clearer insights and more informed conclusions regarding the dataset's characteristics.

Statistics in Mathematics

Statistics is a fundamental branch of mathematics that involves collecting, analyzing, interpreting, and presenting data. Understanding concepts such as joint and marginal relative frequencies is crucial for statistical analysis, providing tools to describe and infer from data. Statistics helps in drawing meaningful conclusions from complex datasets by breaking data down into manageable parts. This is often done using tools like two-way tables, which organize data clearly, making patterns and relationships evident. In statistics, questions about data often revolve around "What does the data tell us?" and "Why are these results significant?". Answers to these questions aid in hypothesis testing, predicting future trends, and making generalizations about larger populations based on sampled data. Key objectives in statistical practice include:

• Ensuring data is accurately measured and reported.
• An understanding of data variability and uncertainty.
• Pursuing methods that lead to reliable conclusions.

Overall, statistics is a powerful tool in understanding the world around us, making it essential for scientific research and practical applications in various fields.