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Chapter 4: Problem 35

The accompanying data on annual maximum wind speed (in meters per second) in Hong Kong for each year in a 45 -year period were given in an article that appeared in the journal Renewable Energy (March 2007). Use the annual maximum wind speed data to construct a boxplot. Is the boxplot approximately symmetric? \(\begin{array}{lllllllll}30.3 & 39.0 & 33.9 & 38.6 & 44.6 & 31.4 & 26.7 & 51.9 & 31.9 \\ 27.2 & 52.9 & 45.8 & 63.3 & 36.0 & 64.0 & 31.4 & 42.2 & 41.1 \\\ 37.0 & 34.4 & 35.5 & 62.2 & 30.3 & 40.0 & 36.0 & 39.4 & 34.4 \\ 28.3 & 39.1 & 55.0 & 35.0 & 28.8 & 25.7 & 62.7 & 32.4 & 31.9 \\ 37.5 & 31.5 & 32.0 & 35.5 & 37.5 & 41.0 & 37.5 & 48.6 & 28.1\end{array}\)

Short Answer

Expert verified
First, compute the necessary statistical values and arrange the data in increasing order. Then, draw a box plot with the calculated minimum, quartile 1, median, quartile 3, and maximum values. By analysing our boxplot, we can conclude if it is symmetric or skewed.

Step by step solution

01

Arrange the data in increasing order

Before we can do anything else, we first need to organize the data in increasing order. This allows us to easily pick out the minimum, maximum and median values.
02

Find the minimum, maximum, median, Q1, and Q3 values

Once we've sorted our data, we can now calculate the necessary statistics for the box plot. These include the minimum and maximum values, which correspond to the smallest and largest numbers in our dataset, respectively. The median is the middle value of our sorted dataset or the average of the two middle numbers if our dataset includes an even number of observations. The first quartile (Q1) is the median of the data points to the left of the median in our sorted data, while the third quartile (Q3) is the median of the data points to the right of the median.
03

Construct the boxplot

Now we are ready to construct our boxplot. The box is formed using Q1, Q3, and the median. The 'whiskers' extend from the ends of the box to the minimum and maximum data points. Any data points beyond these, known as outliers, are marked as individual points.
04

Analyze the boxplot for symmetry

To determine whether our boxplot is symmetric, we check whether or not its two halves (i.e. the values above and below the median) are mirror images of each other. If they are, the boxplot is symmetric. If they are not, the boxplot is skewed (negatively if the longer section is on the left, positively if the longer section is on the right).

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Key Concepts

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

Descriptive Statistics
Descriptive statistics are a fundamental part of understanding data patterns and summaries. They give us a basic overview of our dataset by giving insight into central tendencies, variability, and distribution shape.
In the case of the maximum wind speed data from Hong Kong, descriptive statistics help characterize wind speed patterns over a long period:
  • Minimum and Maximum Values: These represent the smallest and largest wind speed observations in our dataset, helping understand the range.
  • Median: This is the middle value when the data is sorted. It shows the center of the distribution and is less sensitive to extreme values compared to the mean.
  • First Quartile (Q1) and Third Quartile (Q3): These values give us insights into the spread of the middle 50% of our data, also known as the interquartile range (IQR).
By carefully examining these statistics, you gain a clearer picture of how wind speeds vary and can identify key trends or outliers that might affect your analysis.
Quartiles
Quartiles are specific points that divide your dataset into four equal parts. Understanding quartiles is essential when constructing boxplots because they are central to the boxplot's structure. Let's dive deeper into what quartiles tell us:
  • Interpreting the Quartiles: The first quartile (Q1) is the median of the first half of your dataset (excluding the actual median if the number of observations is odd). It shows where 25% of your data falls below.

  • The third quartile (Q3) is the median of the second half, indicating that 75% of your data lies below this point.
  • By calculating Q1 and Q3, you can understand how tightly packed the middle data points are in your dataset.

  • The difference between Q3 and Q1, known as the interquartile range (IQR), represents the spread of the central 50% of your data, essential for identifying potential outliers when plotting a boxplot. Outliers are typically 1.5*IQR below Q1 or above Q3.
By focusing on quartiles, we get insights into the overall distribution of the data, allowing us to understand how it clusters, skews, or spreads out.
Data Visualization
Data visualization, like constructing and analyzing a boxplot, transforms raw data into an easily interpretable format. Boxplots, specifically, provide a summarized visual description of the dataset:
  • Components of a Boxplot: A standard boxplot consists of a rectangle (box) that spans from Q1 to Q3 with a thick line at the median. Lines extending from the box, called whiskers, indicate the range (minimum to maximum excluding outliers).

  • Advantages of Boxplots: Boxplots are particularly useful in showing outliers and the overall spread of the data. They provide a clear visual assessment of the symmetry or skewness of the distribution.

  • Symmetry Analysis: The appearance of the box and whiskers helps us quickly determine symmetry. If the median is central within the box and the whiskers are approximately the same length, the distribution is symmetric. If not, the data might be skewed.
Visualizing data through boxplots allows a comprehensive yet straightforward view of complex datasets, making it easier to communicate findings and identify patterns.

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Most popular questions from this chapter

A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (in seconds) to complete the escape ("Oxygen Consumption and Ventilation During Escape from an Offshore Platform," Ergonomics [1997]: 281-292): \(\begin{array}{lllllllll}389 & 356 & 359 & 363 & 375 & 424 & 325 & 394 & 402 \\\ 373 & 373 & 370 & 364 & 366 & 364 & 325 & 339 & 393 \\ 392 & 369 & 374 & 359 & 356 & 403 & 334 & 397 & \end{array}\) a. Construct a stem-and-leaf display of the data. Will the sample mean or the sample median be larger for this data set? b. Calculate the values of the sample mean and median. c. By how much could the largest time be increased without affecting the value of the sample median? By how much could this value be decreased without affecting the sample median?

The paper "Answer Changing on MultipleChoice Tests" (journal of Experimental Education \([1980]: 18-21)\) reported that for a group of 162 college students, the average number of responses changed from the correct answer to an incorrect answer on a test containing 80 multiple-choice items was \(1.4 .\) The corresponding standard deviation was reported to be \(1.5\). Based on this mean and standard deviation, what can you tell about the shape of the distribution of the variable number of answers changed from right to wrong? What can you say about the number of students who changed at least six answers from correct to incorrect?

In August 2009 , Harris Interactive released the results of the "Great Schools" survey. In this survey, 1086 parents of children attending a public or private school were asked approximately how much time they spent volunteering at school per month over the last school year. For this sample, the mean number of hours per month was \(5.6\) hours and the median number of hours was \(1.0 .\) What does the large difference between the mean and median tell you about this data set?

The San Luis Obispo Telegram-Tribune (October 1,1994 ) reported the following monthly salaries for supervisors from six different counties: \(\$ 5354\) (Kern), \(\$ 5166\) (Monterey), \(\$ 4443\) (Santa Cruz), \(\$ 4129\) (Santa Barbara), \(\$ 2500\) (Placer), and \$2220 (Merced). San Luis Obispo County supervisors are supposed to be paid the average of the two counties among these six in the middle of the salary range. Which measure of center determines this salary, and what is its value? Why is the other measure of center featured in this section not as favorable to these supervisors (although it might appeal to taxpayers)?

In a study investigating the effect of car speed on accident severity, 5000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. For these 5000 accidents, the average speed was \(42 \mathrm{mph}\) and the standard deviation was 15 mph. A histogram revealed that the vehicle speed at impact distribution was approximately normal. a. Roughly what proportion of vehicle speeds were between 27 and \(57 \mathrm{mph} ?\) b. Roughly what proportion of vehicle speeds exceeded \(57 \mathrm{mph} ?\)
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