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. $$ \begin{array}{|l|c|c|c|} \hline & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { Expert } & 62 & 6 & 68 \\ \hline \text { Average } & 275 & 24 & 299 \\ \hline \text { Novice } & 40 & 3 & 43 \\ \hline \text { Total } & 377 & 33 & 410 \\ \hline \end{array} $$

Short answer

The joint relative frequencies for 'Expert', 'Average', and 'Novice' among male are respectively \(\frac{62}{410}\), \(\frac{275}{410}\), and \(\frac{40}{410}\). Those for female are \(\frac{6}{410}\), \(\frac{24}{410}\), and \(\frac{3}{410}\). The marginal relative frequencies for 'Expert', 'Average', 'Novice' are \(\frac{68}{410}\), \(\frac{299}{410}\), \(\frac{43}{410}\), and those for 'Male', 'Female' are \(\frac{377}{410}\), \(\frac{33}{410}\) respectively.

Step-by-step solution

Calculate Joint Relative Frequencies

Start by dividing every cell by the grand total to obtain the joint relative frequencies. That is, divide each of the number of 'Expert', 'Average', and 'Novice' people for both genders by the grand total, which is 410.

Calculate Marginal Relative Frequencies

Next, calculate the marginal relative frequencies, which are the sums of the joint relative frequencies for each row and for each column. So you add up each row and column of the joint relative frequencies computed in step 1 separately.

Create The Two-Way Table

Finally, create a new two-way table, filling it with the joint and marginal relative frequencies computed in the previous steps. This table is a version of the original, only with proportions instead of raw counts.

Key concepts

Joint Relative Frequencies

Joint relative frequencies give us a look into how often two events occur at the same time relative to the total number of occurrences. They're very useful when analyzing data from a two-way table, `like` the one in our example. To find these frequencies, you take each cell in the table (representing specific combinations of categories, like "Male Experts" or "Female Novices") and divide it by the grand total number of observations.

In our case, the grand total is 410, meaning we divide each individual count by 410. For instance, for "Male Experts," we divide 62 by 410. This calculation gives us the joint relative frequency for "Male Experts," which expresses how this particular group compares to everyone in the study. This type of frequency lets researchers understand the relationship between two categorical variables in a dataset visually and numerically.

Calculating these frequencies is crucial because they simplify the data, allowing us to see percentages and proportions instead of raw numbers. These insights aid in comparing how different categories interact and influence one another.

Marginal Relative Frequencies

Marginal relative frequencies focus on the totals of rows and columns in a two-way table. Unlike joint relative frequencies, which focus on intersections, marginal frequencies look at one category at a time. In simple terms, they show how often something happens within a specific group, compared to the overall dataset.

To calculate these, we sum up the joint relative frequencies in each row or column, which was found in the previous step. For example, the marginal relative frequency for males is found by adding the joint relative frequencies calculated for all male categories - "Male Experts," "Male Average," and "Male Novice."

Similarly, we do this for each column to understand how much each individual factor contributes to the whole set. Marginal frequencies help in understanding the general importance or weight of a single category, like gender or expert level, in the dataset. They act as a summary, revealing overarching trends that might not be visible from looking at individual entries.

Statistics

Statistics is a vast field that helps us make sense of data and draw conclusions from it. It involves collecting, analyzing, interpreting, presenting, and organizing data. In our example, we used statistics to analyze a two-way frequency table. This table is a useful tool in statistics for exploring relationships between categorical variables.

Using statistical methods, we converted raw data into joint and marginal relative frequencies, making the information easier to understand and interpret. This transformation provides clarity, which numbers in a data set might not immediately offer. Statistics allows us to not only describe data but also to infer or predict patterns and trends.

It's essential because it converts complex, voluminous data into meaningful outcomes, helping researchers, businesses, and individuals make informed decisions based on numerical evidence. Understanding and applying statistical principles enables us to draw reliable conclusions and make well-informed predictions about future events based on current data.