Question

A recent study shows that antibiotics added to animal feed to accelerate growth can become airborne. Some of these drugs can be toxic if inhaled and may increase the evolution of antibiotic-resistant bacteria. Scientists \(^{9}\) analyzed 20 samples of dust particles from animal farms. Tylosin, an antibiotic used in animal feed that is chemically related to erythromycin, showed up in 16 of the samples. (a) What is the variable in this study? What are the individual cases? (b) Display the results in a frequency table. (c) Make a bar chart of the data. (d) Give a relative frequency table of the data.

Short answer

The variable here is the presence of antibiotic 'Tylosin'. The individual cases are the 20 samples collected from farms. Frequency table represents 16 presence samples and 4 absence samples. The bar chart would have two bars of heights 16 and 4 for presence and absence respectively. The relative frequency table shows 0.8 for presence and 0.2 for absence.

Step-by-step solution

Understanding the Variable and Individual Cases

The variable in this study is the presence of the antibiotic 'Tylosin' in the dust particles from animal farms. The individual cases are the 20 samples of dust particles collected from the farms.

Creating the Frequency Table

A frequency table can be created by listing out the possible outcomes (presence or absence of Tylosin), and corresponding frequencies (number of times each outcome occurred). In this case, Tylosin was present in 16 samples and absent in 4 samples.

Plotting a Bar Chart

Make a bar chart with two bars, one for 'Present' and the other for 'Absent'. The height of each bar corresponds to the number of samples for each category. The 'Present' bar would be higher than the 'Absent' bar, reflecting the fact that Tylosin was found in more samples

Creating a relative Frequency table

To create a relative frequency table, divide the frequencies by the total number of samples. Here, the total number of samples is 20. So, the relative frequency of 'Present' would be 16/20 = 0.8 and the relative frequency of 'Absent' would be 4/20 = 0.2

Key concepts

Variables in Studies

When we talk about 'variables in studies', we are referring to the different elements that we can measure or observe in research. Variables are the backbone of any study, as they help to quantify and analyze the changes or differences that are of interest to the researcher. In the study mentioned, the variable of interest is the presence of the antibiotic Tylosin in dust particles on animal farms. These variables can be either qualitative (categorized by descriptive terms) or quantitative (measured numerically).

Each sample collected and analyzed for Tylosin is considered an individual case. In studies, individual cases are the subjects or items that you are studying, and each contributes a single piece of data to the overall study. For example, in the exercise, each of the 20 dust samples from different animal farms serves as an individual case for the study. This helps researchers in determining the frequency of the presence of Tylosin across the sampled farms.

Frequency Table

A frequency table is a simple but powerful tool in statistics that summarizes how often each value in a set of data occurs. It provides a clear visual summary that helps researchers and students to understand the distribution of data. Constructing a frequency table involves tallying up the number of times each value appears. For the given exercise, values are either 'Present' (Tylosin found) or 'Absent' (Tylosin not found).

To create this table, you list the possible outcomes and then count how many times each outcome is observed. This can be done by creating two columns: one for the value (in this case, 'Present' or 'Absent') and one for the frequency. The textbook solution highlights that Tylosin was 'Present' in 16 out of the 20 samples and 'Absent' in the remaining four—these numbers form the basis of the frequency table.

Bar Chart

A bar chart is a graphical representation that uses bars of different heights or lengths to show the frequencies of various categories or values. It's a highly effective visual tool for displaying and comparing quantitative information. When constructing a bar chart, each category is represented by a bar. The length or height of the bar corresponds to the count or frequency of that category.

In the context of our exercise, a bar chart helps visually summarize the presence of Tylosin in the dust samples. The two bars would represent 'Present' and 'Absent' accordingly. The height of the 'Present' bar would be proportionate to the number of samples with Tylosin (16 samples), while the 'Absent' bar's height corresponds to the samples without Tylosin (4 samples). This visual representation helps students quickly grasp the prevalence of Tylosin in the analyzed samples.

Relative Frequency Table

While a frequency table shows the count of occurrences, a relative frequency table shows the proportion or percentage of times each value occurs relative to the total number of occurrences. To create a relative frequency table, each frequency is divided by the total number of data points. In the given exercise, the relative frequency for the 'Present' category would be computed by dividing 16 (the number of 'Present' samples) by the total of 20 samples, resulting in 0.8 or 80%. Similarly, the relative frequency for the 'Absent' category would be 4 divided by 20, yielding 0.2 or 20%.

This conversion to percentages can be easier to understand for many students, as it allows for a quick comparison of the different categories and shows the proportion of each category in the context of the total dataset. Including relative frequencies alongside frequencies can significantly enhance the interpretability of the data for students.