Question

In a survey, 49 people received a flu vaccine before the flu season and 63 people did not receive the vaccine. Of those who receive the flu vaccine, 16 people got the flu. Of those who did not receive the vaccine, 17 got the flu. Make a two-way table that shows the joint and marginal relative frequencies.

Short answer

The two-way table with joint relative frequencies is as follows: 'Received flu vaccine and got sick' = 0.14, 'Received flu vaccine and didn't get sick' = 0.29, 'Didn't receive flu vaccine and got sick' = 0.15, 'Didn't receive flu vaccine and didn't get sick' = 0.41. The marginal relative frequencies are: 'Received flu vaccine' = 0.43, 'Didn't receive flu vaccine' = 0.56, 'Got sick' = 0.29, 'Didn't get sick' = 0.71.

Step-by-step solution

Calculate total number of people in the survey

Add the number of people who received the vaccine and those who did not. Total = 49 (received vaccine) + 63 (didn't receive vaccine) = 112.

Fill in the joint frequencies

Fill in the four respective fields with the number of people who got the flu and who didn't, both for those who received and didn't receive the vaccine.

Calculate marginal frequencies

Add the joint frequencies by row and column to get marginal frequencies. For instance, the number of people who received the vaccine and got the flu plus those who received the vaccine and didn't get the flu will give the marginal frequency for the 'received vaccine' row.

Calculate the joint relative frequencies

Divide each joint frequency by the total number of people (112). Record these in the table under the respective fields.

Calculate the marginal relative frequencies

Add joint relative frequencies of each row and column to get the marginal relative frequencies. They should be placed in the marginal frequency row and column.

Key concepts

Understanding Joint Frequencies

Joint frequencies in a two-way table represent the count of occurrences across two distinct categories. It’s about finding how often two specific events happen together. In our survey example:

• 16 people received the vaccine and got the flu.
• 33 people received the vaccine and did not get the flu.
• 17 people did not receive the vaccine and got the flu.
• 46 people did not receive the vaccine and did not get the flu.

Together, these numbers fill the interior cells of the table.
These help us quickly see the connection between receiving the vaccine and contracting the flu. By breaking data into smaller segments, joint frequencies provide a detailed insight into relationships from the survey.

Exploring Marginal Frequencies

Marginal frequencies are the totals of each row and column in a two-way table. They are like summaries for one category, neglecting the other. In the survey, we calculated:

• 49 people in total received the vaccine (16 got the flu, 33 didn't).
• 63 people in total did not receive the vaccine (17 got the flu, 46 didn't).

These numbers appear at the end of each row and column. They provide a snapshot of the total for one group regardless of other factors.
By employing marginal frequencies, we understand the broader category dynamics, such as how many people were vaccinated overall and how many got the flu regardless of vaccination.

The Role of Relative Frequencies

Relative frequencies offer a proportion-based perspective of the data, showing the frequency of events as part of a whole. To calculate these, we divide each joint frequency by the total number of people surveyed, which is 112 in our example.
For instance:

• The relative frequency of those vaccinated and getting the flu is \(\frac{16}{112} \approx 0.143\).
• For those vaccinated and not getting the flu, it's \(\frac{33}{112} \approx 0.295\).

Relative frequencies are placed within the same cells as the joint frequencies but can also be summed at the margin to find marginal relative frequencies.
This approach offers a percentage view and helps in comparing the likelihood of events, providing clarity on how significant the numbers truly are.

Survey Data Analysis in Context

Analyzing survey data through a two-way table is a powerful method to understand complex relationships. It helps in logically arranging the data for easy interpretation.

• We start by setting up joint and marginal frequencies to grasp the overall trends swiftly.
• Relative frequencies then refine this understanding by expressing these occurrences in terms of proportions of the entire survey group.

Such tables also simplify the steps of presenting and analyzing intricate data sets concisely. They serve well in decisions or conclusions that require examining relationships between different variables, like determining the effectiveness of flu vaccines from our survey.
In the end, survey data analysis using two-way tables proves to be an invaluable skill in extracting meaningful insights and supporting informed decision-making.