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Chapter 3: Problem 17

The data show the heights (in feet) of the 10 largest dams in the United States. Identify the five-number summary and the interquartile range, and draw a boxplot. \(\begin{array}{llllllllll}770 & 730 & 717 & 710 & 645 & 606 & 602 & 585 & 578 & 564\end{array}\)

Short Answer

Expert verified
Five-number summary: 564, 593.5, 625.5, 713.5, 770; IQR: 120; A boxplot is drawn using these values.

Step by step solution

01

Order the Data Set

First, we need to arrange the given data in ascending order to identify the five-number summary easily. The data in ascending order is: 564, 578, 585, 602, 606, 645, 710, 717, 730, 770.
02

Identify the Five-Number Summary

The five-number summary consists of five key data points: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.- Minimum: 564- Maximum: 770- Median (Q2): Middle value of the ordered list, for an even number of elements (n=10), median is the average of the 5th and 6th values: \( \frac{606 + 645}{2} = 625.5 \)- First Quartile (Q1): Median of the first half of the data, more specifically between the 1st and 5th element: \( \frac{585 + 602}{2} = 593.5 \)- Third Quartile (Q3): Median of the second half of the data, specifically between the 6th and 10th element: \( \frac{710 + 717}{2} = 713.5 \)
03

Calculate the Interquartile Range (IQR)

The interquartile range (IQR) is calculated by subtracting the first quartile from the third quartile. \[ IQR = Q3 - Q1 = 713.5 - 593.5 = 120 \]
04

Draw the Boxplot

To draw the boxplot: 1. Draw a number line that covers the range of the data (from 560 to 780). 2. Mark the values of the five-number summary along the number line. 3. Draw a rectangular box from Q1 to Q3. 4. Draw a vertical line at the median (Q2) inside this box. 5. Extend the 'whiskers' from the box to the minimum and maximum data points.

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Key Concepts

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

Understanding the Five-Number Summary
The five-number summary is a simple yet powerful way to describe a dataset. It's composed of five specific points: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Think of it like a snapshot that gives you a good sense of the data's spread and distribution.
Let's break these down one by one:
  • **Minimum**: The smallest value in the dataset.
  • **Maximum**: The largest value in the dataset.
  • **Median**: The middle value that divides the dataset into two equal halves. For even-numbered data, like in this exercise, the median is the average of the two central numbers.
  • **First Quartile (Q1)**: The median of the first half of the data, indicating that 25% of the data is below this value.
  • **Third Quartile (Q3)**: The median of the second half of the data, showing that 75% of the data is below this value.
By identifying these points, we get a quick overview of the dataset's range and how data points are spread around the median. In our example with the heights of the largest dams, the summary highlights values from 564 to 770 feet, showing a substantial spread.
Delving into the Interquartile Range
The Interquartile Range (IQR) is a crucial statistic that captures the middle 50% of a dataset, highlighting the range where the majority of values fall.To calculate the IQR, you simply subtract the first quartile (Q1) from the third quartile (Q3). It is given by:\[IQR = Q3 - Q1\]In our exercise, the values were:
  • Q1 = 593.5 feet
  • Q3 = 713.5 feet
Thus, the IQR calculated was 120 feet.This value holds practical significance as it shows where the most consistent data points are located, filtering out potential outliers. A smaller IQR indicates a more tightly clustered dataset, while a larger IQR shows more spread. In our dataset of dam heights, it tells us there's considerable variation within this interquartile range, which can be relevant when assessing the dataset's variability.
The Role of Statistics Education
Statistics education equips students with tools to make sense of data in everyday situations. Key concepts like the five-number summary and interquartile range are foundational in understanding how data is distributed and analyzed.
  • **Development of Analytical Skills**: Understanding statistical tools and summarizations like boxplots help in making informed decisions based on data interpretation.
  • **Problem-Solving Abilities**: Students learn to compute key statistics manually and through software, enhancing problem-solving techniques useful in multiple fields.
  • **Critical Thinking**: Gaining insights from data summaries teaches students to question and analyze findings, fostering critical thinking.
These skills prepare learners not only to excel in academic contexts but also to apply them in real-world scenarios. In complex data-driven environments, such as economics or scientific research, having a solid statistical foundation is invaluable. Moreover, it's instrumental in developing a numerate society capable of digesting vast amounts of data effectively.

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Most popular questions from this chapter

Describe which measure of central tendency \(-\) mean, median, or mode-was probably used in each situation. a. One-half of the factory workers make more than \(\$ 5.37\) per hour, and one- half make less than \(\$ 5.37\) per hour. b. The average number of children per family in the Plaza Heights Complex is 1.8 . c. Most people prefer red convertibles over any other color. d. The average person cuts the lawn once a week. e. The most common fear today is fear of speaking in public. f. The average age of college professors is 42.3 years.

Harmonic Mean The harmonic mean (HM) is defined as the number of values divided by the sum of the reciprocals of each value. The formula is $$\mathrm{HM}=\frac{n}{\Sigma(1 / X)}$$ For example, the harmonic mean of \(1,4,5,\) and 2 is $$\mathrm{HM}=\frac{4}{1 / 1+1 / 4+1 / 5+1 / 2} \approx 2.051$$ This mean is useful for finding the average speed. Suppose a person drove 100 miles at 40 miles per hour and returned driving 50 miles per hour. The average miles per hour is not 45 miles per hour, which is found by adding 40 and 50 and dividing by 2 . The average is found as shown. Since Time \(=\) distance \(\div\) rate then Time \(1=\frac{100}{40}=2.5\) hours to make the trip Time \(2=\frac{100}{50}=2\) hours to return Hence, the total time is 4.5 hours, and the total miles driven are \(200 .\) Now, the average speed is $$\text { Rate }=\frac{\text { distance }}{\text { time }}=\frac{200}{4.5} \approx 44.444 \text { miles per hour }$$ This value can also be found by using the harmonic mean formula $$\mathrm{HM}=\frac{2}{1 / 40+1 / 50} \approx 44.444$$ Using the harmonic mean, find each of these. a. A salesperson drives 300 miles round trip at 30 miles per hour going to Chicago and 45 miles per hour returning home. Find the average miles per hour. b. A bus driver drives the 50 miles to West Chester at 40 miles per hour and returns driving 25 miles per hour. Find the average miles per hour. c. A carpenter buys \(\$ 500\) worth of nails at \(\$ 50\) per pound and \(\$ 500\) worth of nails at \(\$ 10\) per pound. Find the average cost of 1 pound of nails.

The geometric mean (GM) is defined as the \(n\) th root of the product of \(n\) values. The formula is $$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$ The geometric mean of 4 and 16 is $$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$ The geometric mean of \(1,3,\) and 9 is $$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$ The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a \(20 \%\) raise after 1 year of service and a \(10 \%\) raise after the second year of service, the average percentage raise per year is not 15 but \(14.89 \%,\) as shown. $$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$ Or $$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$ His salary is \(120 \%\) at the end of the first year and \(110 \%\) at the end of the second year. This is equivalent to an average of \(14.89 \%\), since \(114.89 \%-100 \%=\) \(14.89 \% .\) This answer can also be shown by assuming that the person makes \(\$ 10,000\) to start and receives two raises of \(20 \%\) and \(10 \%\). $$\begin{array}{l}\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000 \\\\\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200\end{array}$$ Find the geometric mean of each of these. a. The growth rates of the Living Life Insurance Corporation for the past 3 years were \(35,24,\) and \(18 \%\). b. A person received these percentage raises in salary over a 4-year period: \(8,6,4,\) and \(5 \%\). c. A stock increased each year for 5 years at these percentages: \(10,8,12,9,\) and \(3 \%\). d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were \(1,3,\) and \(5.5 \% .\)

The lengths (in feet) of the main span of the longest suspension bridges in the United States and the rest of the world are shown below. Which set of data is more variable? United States 4205,4200,3800,3500,3478,2800,2800,2310 World \(\quad 6570,5538,5328,4888,4626,4544,4518,3970\)

The data show the sizes in square miles of notable islands in the Baltic Sea and the Aleutian Islands. Construct a boxplot for each data set and compare the distributions. $$ \begin{array}{lc} \text { Baltic Sea } & \text { Aleutian Islands } \\ \hline 610 & 275 \\ 228 & 1051 \\ 1154 & 571 \\ 1159 & 686 \\ 2772 & 350 \end{array} $$
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