91Ó°ÊÓ

Americium \(241\left({ }^{241} \mathrm{Am}\right)\) is a radioactive material used in the manufacture of smoke detectors. The article "Retention and Dosimetry of Injected \({ }^{241}\) Am in Beagles" (Radiation Research [1984]: \(564-575\) ) described a study in which 55 beagles were injected with a dose of \({ }^{241} \mathrm{Am}\) (proportional to each animal's weight). Skeletal retention of \({ }^{241} \mathrm{Am}\) (in microcuries per kilogram) was recorded for each beagle, resulting in the following data: \(\begin{array}{lllllll}0.196 & 0.451 & 0.498 & 0.411 & 0.324 & 0.190 & 0.489 \\\ 0.300 & 0.346 & 0.448 & 0.188 & 0.399 & 0.305 & 0.304 \\ 0.287 & 0.243 & 0.334 & 0.299 & 0.292 & 0.419 & 0.236 \\ 0.315 & 0.447 & 0.585 & 0.291 & 0.186 & 0.393 & 0.419\end{array}\) \(\begin{array}{lllllll}0.335 & 0.332 & 0.292 & 0.375 & 0.349 & 0.324 & 0.301 \\ 0.333 & 0.408 & 0.399 & 0.303 & 0.318 & 0.468 & 0.441 \\ 0.306 & 0.367 & 0.345 & 0.428 & 0.345 & 0.412 & 0.337 \\\ 0.353 & 0.357 & 0.320 & 0.354 & 0.361 & 0.329 & \end{array}\) a. Construct a frequency distribution for these data, and draw the corresponding histogram. b. Write a short description of the important features of the shape of the histogram.

Step by step solution

01

Frequency distribution

The initial step is to organize the given data set into a sensible frequency distribution or count occurrence of each data point. Sorting the data can be helpful. Compute the frequency for each data point and arrange this as a table.

02

Construction of Histogram

After arranging the data into a frequency distribution, next is to draw a histogram. The x-axis (horizontal) should represent the range of values in the data set, while the y-axis (vertical) should represent the frequency of these values. Choose suitable intervals for the x-axis. Every interval should have a vertical bar whose height corresponds to how often values in the interval occur in the data set. Construct the bars on the intervals.

03

Analysis of Histogram

Upon completion of the histogram, analyze the shape. This includes identification of any symmetry, skewness, outliers, and the peak. Notice the regions where the histogram bulges, its tails, and whether or not it’s symmetrical. Observe if it’s skewed to the left or right, and if there are any unusual gaps or peaks.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Histogram Construction

Creating a histogram is a visual technique used to represent frequency distributions. The first step involves organizing the given dataset. In this case, the data set consists of skeletal retention of Americium-241 in beagles. Start by sorting the data from the smallest to the largest value. Then, group these data points into intervals or "bins." The width of these bins should be chosen carefully, balancing between too much and too little detail.
Once the data is grouped, the next step is to draw the histogram. The x-axis (horizontal) will display the intervals of the data values, and the y-axis (vertical) will show the frequency of data within each bin. Each interval gets a bar, and the height of the bar corresponds to how many data points fall within that bin.
This method allows us to see at a glance where the majority of data points fall and how they are distributed across the range. Whether tackling homework or real data analysis tasks, knowing how to construct a histogram is a must-have skill.

Data Analysis

Analyzing data involves extracting useful information from it. Once a histogram is constructed, we can dive into more detailed analysis to understand the data's story.
Key aspects to consider when analyzing the histogram include the central tendency, spread, and any patterns or anomalies. Questions to ask might include: Where does the peak occur? Which intervals have the highest or lowest frequencies? Is the distribution uniform or does it bulge or taper in certain areas? All these observations can reveal critical insights into the characteristics of the data.
Moreover, analyzing how the data is skewed (left or right) can suggest underlying trends or external factors affecting the data. Unusual gaps or exceptional peaks can also indicate errors in data collection or unique phenomena worth investigating further.

Statistical Graphs

Statistical graphs are an essential part of data analysis, enhancing our understanding by presenting numerical data visually. In the context of this exercise, the histogram is a type of statistical graph that helps visualize frequency distribution.
Understanding statistical graphs like histograms not only makes the data more approachable but also aids in identifying relationships among variables. They make complex data sets comprehensible through instantly recognizable patterns and shapes.
Beyond histograms, other statistical graphs such as bar charts, pie charts, or scatterplots can be used to illustrate different aspects of data. Each type of graph provides unique insights and is suited to particular data types or analytical purposes. Mastering these techniques enhances your ability to effectively interpret data and communicate findings in any field involving statistical analysis.