Question

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.

Short answer

The histogram will provide a visual presentation of the distribution of the skeletal retention of Amercium-241 in the tested beagles. You can assess the symmetry, skewness, potential outliers and peak of the data set from the histogram. However, without actual data computation and graphical representation, definitive assessment is not feasible.

Step-by-step solution

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.

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.

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.