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Chapter 1: Problem 3

Of great importance to residents of central Florida is the amount of radioactive material present in the soil of reclaimed phosphate mining areas. Measurements of the amount of \(^{238} \mathrm{U}\) in 25 soil samples were as follows (measurements in picocuriespergram):$$\begin{array}{rrrrr}.74 & 6.47 & 1.90 & 2.69 & .75 \\\\.32 & 9.99 & 1.77 & 2.41 & 1.96 \\\1.66 & .70 & 2.42 & .54 &3.36 \\\3.59 & .37 & 1.09 & 8.32 & 4.06 \\ 4.55 & .76 & 2.03 & 5.70 & 12.48\end{array}$$.Construct a relative frequency histogram for these data.

Short Answer

Expert verified
Create a relative frequency histogram with suitable bin intervals and heights.

Step by step solution

01

Organize the Data into Bins

To construct a relative frequency histogram, we'll first need to determine the range of our data and then select suitable bins. The smallest value is 0.32, and the largest value is 12.48. We'll create intervals that can neatly encompass all values. A suitable option might be to use bins of width 2: 0-2, 2-4, 4-6, 6-8, 8-10, and 10-12.
02

Count the Frequency of Data in Each Bin

Go through the data, and for each observation, determine which bin it falls into: - 0-2: 11 samples - 2-4: 7 samples - 4-6: 3 samples - 6-8: 1 sample - 8-10: 2 samples - 10-12: 1 sample These numbers represent the frequency of observations in each bin.
03

Compute the Relative Frequencies

The relative frequency is the frequency for each bin divided by the total number of observations. We have 25 total samples, so divide each frequency by 25:- 0-2: \(\frac{11}{25} = 0.44\)- 2-4: \(\frac{7}{25} = 0.28\)- 4-6: \(\frac{3}{25} = 0.12\)- 6-8: \(\frac{1}{25} = 0.04\)- 8-10: \(\frac{2}{25} = 0.08\)- 10-12: \(\frac{1}{25} = 0.04\)
04

Construct the Histogram

Draw the X-axis to represent the bin intervals and the Y-axis to represent the relative frequencies. For each bin, draw a bar up to the height of its relative frequency. Ensure that the width of each bar corresponds to the bin width (which is 2 in this case). - For 0-2, draw a bar up to 0.44 - For 2-4, draw a bar up to 0.28 - For 4-6, draw a bar up to 0.12 - For 6-8, draw a bar up to 0.04 - For 8-10, draw a bar up to 0.08 - For 10-12, draw a bar up to 0.04

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Key Concepts

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

Data Binning
Data binning is an important step in creating a histogram. It involves organizing raw data into groups called bins. This helps simplify and present data more clearly. Each bin contains a range of data points.
In our example concerning radioactive material measurements in soil, the data ranges from 0.32 to 12.48 picocuries per gram. We chose bins of width 2, like 0-2, 2-4, etc. This setup helps in visually summarizing the data.
Choosing the right bin width is crucial. If bins are too small, the histogram becomes cluttered, showing too much detail. If too large, important details may be lost. A balance helps in identifying patterns or trends. For our data, a width of 2 was appropriate to provide a clear view of how measurements were distributed.
Frequency Distribution
Once the data is organized into bins, the next step is to calculate the frequency distribution. This indicates how often data points fall within each bin.
For the radioactive material data, we counted how many samples fell into each chosen bin, such as 0-2, 2-4, and so forth. For instance, 11 samples were between 0 and 2 picocuries per gram.
The process typically involves counting the number of occurrences in each bin, giving a straightforward view of data concentration within different ranges. Frequency distribution helps highlight key aspects of the data, such as where most values cluster or how they spread over different intervals.
  • 0-2: 11 samples
  • 2-4: 7 samples
  • 4-6: 3 samples
  • 6-8: 1 sample
  • 8-10: 2 samples
  • 10-12: 1 sample
Radioactive Material Measurement
Measurement of radioactive materials, like uranium in soil, is crucial for safety and environmental health. In the exercise, uranium levels in Central Florida soil samples were measured.
Understanding these measurements helps to gauge the potential risk or safety of an area. By examining the concentration of radioactive material, experts can decide on necessary actions.
The measurements in the presented case were given in picocuries per gram, a unit used to measure radioactivity. Tools and instruments used for such measures must be precise, as small errors can lead to significant misunderstandings about safety. Soil measurement, as shown in the data, varies greatly, with values ranging from as low as 0.32 to as high as 12.48 picocuries per gram.
The frequency information provides insight into the spread of radioactive material across different sites. Areas with higher readings may require more detailed investigation or remedial actions to ensure they are safe.

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Most popular questions from this chapter

A machine produces bearings with mean diameter 3.00 inches and standard deviation 0.01 inch. Bearings with diameters in excess of 3.02 inches or less than 2.98 inches will fail to meet quality specifications. a. Approximately what fraction of this machine's production will fail to meet specifications? b. What assumptions did you make concerning the distribution of bearing diameters in order to answer this question?

Prove that the sum of the deviations of a set of measurements about their mean is equal to zero; that is, $$\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)=0$$

Let \(k \geq 1\). Show that, for any set of \(n\) measurements, the fraction included in the interval \(\bar{y}-k s\) to \(\bar{y}+k s\) is at least \(\left(1-1 / k^{2}\right) .\) [Hint: $$s^{2}=\frac{1}{n-1}\left[\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}\right]$$ In this expression, replace all deviations for which \(\left|y_{i}-\bar{y}\right| \geq k s\) with \(k s\). Simplify.] This result is known as Tchebysheff's theorem. \(^{*}\)

Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who desire a low-impact aerobics program. In a study to investigate the relationship between exercise cadence and heart rate, the heart rates of 20 healthy volunteers were measured at a cadence of 48 cycles per minute (a cycle consisted of two steps). The data are as follows: $$\begin{array}{rrrrrrrrrr} 87 & 109 & 79 & 80 & 96 & 95 & 90 & 92 & 96 & 98 \\ 101 & 91 & 78 & 112 & 94 & 98 & 94 & 107 & 81 & 96 \end{array}$$ a. Use the range of the measurements to obtain an estimate of the standard deviation. b. Construct a frequency histogram for the data. Use the histogram to obtain a visual approximation to \(\bar{y}\) and \(s\) c. Calculate \(\bar{y}\) and \(s .\) Compare these results with the calculation checks provided by parts \((\mathrm{a})\) and (b). d. Construct the intervals \(\bar{y} \pm k s, k=1,2,\) and \(3,\) and count the number of measurements falling in each interval. Compare the fractions falling in the intervals with the fractions that you would expect according to the empirical rule.

Resting breathing rates for college-age students are approximately normally distributed with mean 12 and standard deviation 2.3 breaths per minute. What fraction of all college-age students have breathing rates in the following intervals? a. 9.7 to 14.3 breaths per minute b. 7.4 to 16.6 breaths per minute c. 9.7 to 16.6 breaths per minute d. Less than 5.1 or more than 18.9 breaths per minute
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