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Use the given data to construct a boxplot and identify the 5-number summary. Radiation in Baby Teeth Listed below are amounts of strontium- 90 (in millibecquerels, or mBq) in a simple random sample of baby teeth obtained from Pennsylvania residents born after 1979 (based on data from "An Unexpected Rise in Strontium-90 in U.S. Deciduous Teeth in the 1990s;" by Mangano et. al., Science of the Total Environment). \begin{tabular}{|lllllllllll} 128 & 130 & 133 & 137 & 138 & 142 & 142 & 144 & 147 & 149 & 151 & 151 & 151 & 155 \end{tabular} " \(\begin{array}{rrrrr}156 & 161 & 163 & 163 & 166 & 172\end{array}\)

Short Answer

Expert verified
The 5-number summary is: Min = 128, Q1 = 140, Median = 150, Q3 = 155.5, Max = 172.

Step by step solution

01

Organize the Data

Begin by arranging the given data set in ascending order: 128, 130, 133, 137, 138, 142, 142, 144, 147, 149, 151, 151, 151, 155, 156, 161, 163, 163, 166, 172
02

Identify the Minimum and Maximum Values

The minimum value in the dataset is 128 and the maximum value is 172. These form the lower and upper extremes of the 5-number summary.
03

Find the Median (Q2)

To find the median, locate the middle number in the sorted dataset. With an even number of observations (20), the median is the average of the 10th and 11th values: \(\frac{149 + 151}{2} = 150\). The median (Q2) is therefore 150.
04

Calculate the First Quartile (Q1)

To locate the first quartile, compute the median of the lower half of the data. The 1st quartile (Q1) is the median of the first 10 numbers: \(\frac{138 + 142}{2} = 140\). Therefore, Q1 is 140.
05

Calculate the Third Quartile (Q3)

To locate the third quartile, compute the median of the upper half of the data. The 3rd quartile (Q3) is the median of the last 10 numbers: \(\frac{155 + 156}{2} = 155.5\). Therefore, Q3 is 155.5.
06

Construct the Boxplot

Using the 5-number summary (Minimum, Q1, Median, Q3, Maximum), construct a boxplot. Mark the minimum (128), Q1 (140), median (150), Q3 (155.5), and maximum (172) on a number line. Draw a box from Q1 to Q3 and a line inside the box at the median. Extend whiskers from the box to the minimum and maximum.

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Key Concepts

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

five-number summary
To create a boxplot, it’s essential to understand the 'five-number summary'. The five-number summary comprises five key descriptive statistics of a dataset: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and the maximum. These metrics define specific data points that help summarize the distribution. The minimum represents the smallest number in the set, and the maximum is the largest. The median (Q2) is the middle value, dividing the dataset into two equal halves. The first quartile (Q1) is the median of the lower half, and the third quartile (Q3) is the median of the upper half. Together, these five numbers provide a snapshot of data distribution and variability.
quartiles
Quartiles are critical components of the five-number summary that split a dataset into four equal parts. The three quartiles are Q1, Q2 (the median), and Q3. Both Q1 and Q3 are medians of their respective halves of the data. For example, with an ordered dataset like 128, 130, 133, 137, 138, 142, 142, 144, 147, 149, 151, 151, 151, 155, 156, 161, 163, 163, 166, 172, Q1 is the median of 128 to 149, which is 140. The third quartile, Q3, is the median of 151 to 172, calculated to be 155.5. Quartiles thus help in understanding the data spread and identifying the middle values of segments within the dataset.
descriptive statistics
Descriptive statistics provide simple summaries about the sample and the measures. They give insights into the dataset with basic information like mean, median, mode, variance, and standard deviation. In our context, the five-number summary (minimum, Q1, median, Q3, maximum) is among these descriptive measures. These metrics help students and researchers understand the data's core attributes without diving into detailed analysis. For instance, the median shows the data's central tendency, while the quartiles (Q1 and Q3) reveal data distribution patterns. Descriptive statistics lay the foundation for further statistical interpretation and analysis.
boxplot interpretation
Boxplots, or box-and-whisker plots, are graphical representations constructed from the five-number summary. They offer a visual summary of data distribution. In a boxplot, a box is drawn from Q1 to Q3, indicating the interquartile range (IQR). A line inside the box marks the median (Q2). Whiskers extend from the box to the minimum and maximum values. This visualization helps in quickly identifying data spread, central values, and potential outliers. For example, a boxplot for our data set ranging from 128 to 172 will show most data points lying between Q1 (140) and Q3 (155.5). The length of the box shows data concentration and variability within the interquartile range. Boxplots are excellent tools for comparing multiple datasets and observing their distribution patterns visually.

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Most popular questions from this chapter

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