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Chapter 3: Problem 99

Prepare a box-and-whisker plot for the following data: \(\begin{array}{llllllll}36 & 43 & 28 & 52 & 41 & 59 & 47 & 61 \\ 24 & 55 & 63 & 73 & 32 & 25 & 35 & 49 \\ 31 & 22 & 61 & 42 & 58 & 65 & 98 & 34\end{array}\) Does this data set contain any outliers?

Short Answer

Expert verified
The box-and-whisker plot can be made for the provided data set using the identified five number summary. The data set does not contain any outliers.

Step by step solution

01

Organize the Data

First sort the data from smallest to largest. This becomes 22, 24, 25, 28, 31, 32, 34, 35, 36, 41, 42, 43, 47, 49, 52, 55, 58, 59, 61, 61, 63, 65, 73, 98.
02

Determine the Five Number Summary

Next, find the smallest number (22), lower quartile \(Q1 = 32.5\) (the median of the lower half of the data), median \(Q2 = 43.5\), upper quartile \(Q3 = 61\) (the median of the upper half of the data), and the highest number (98).
03

Make the Box-Whisker Plot

Start by putting a scale on a number line that includes the smallest and largest numbers in the data set. Then draw a box that starts at Q1 and ends at Q3. Draw an additional line (the 'whisker') from the smallest number to Q1, and another line from Q3 to the largest number. Put a line in the box at Q2 to represent the median.
04

Determine if there are outliers

Find the inner quartile range (IQR), which equals Q3 - Q1 = 61 - 32.5 = 28.5. To identify outliers, multiply IQR by 1.5 and then subtract this value from Q1 (32.5 - 1.5*28.5 = -10) and add it to Q3 (61 + 1.5*28.5 = 103.75). Therefore, anything below -10 and above 103.75 would be considered outliers. In this case, the data set does not contain any outliers because all the numbers fall within this range.

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Most popular questions from this chapter

The heights of five starting players on a basketball team have a mean of 76 inches, a median of 78 inches, and a range of 11 inches. a. If the tallest of these five players is replaced by a substitute who is 2 inches taller, find the new mean, median, and range. b. If the tallest player is replaced by a substitute who is 4 inches shorter, which of the new values (mean, median, range) could you determine, and what would their new values be?

The trimmed mean is calculated by dropping a certain percentage of values from each end of a ranked data set. The trimmed mean is especially useful as a measure of central tendency when a data set contains a few outliers. Suppose the following data give the ages (in years) of 10 employees of a company: \(\begin{array}{llllll}47 & 53 & 38 & 26 & 39 & 49\end{array}\) 19 \(\begin{array}{ccc}67 & 31 & 23\end{array}\) To calculate the \(10 \%\) trimmed mean, first rank these data values in increasing order; then drop \(10 \%\) of the smallest values and \(10 \%\) of the largest values. The mean of the remaining \(80 \%\) of the values will give the \(10 \%\) trimmed mean. Note that this data set contains 10 values, and \(10 \%\) of 10 is 1 . Thus, if we drop the smallest value and the largest value from this data set, the mean of the remaining 8 values will be called the \(10 \%\) trimmed mean. Calculate the \(10 \%\) trimmed mean for this data set.

Which of the three measures of central tendency (the mean, the median, and the mode) can assume more than one value for a data set? Give an example of a data set for which this summary measure assumes more than one value.

The following data set belongs to a population: 5 \(\begin{array}{llllll}2 & 0 & -9 & 16 & 10 & 7\end{array}\)

The following stem-and-leaf diagram gives the distances (in thousands of miles) driven during the past year by a sample of drivers in a city. $$ \begin{array}{l|llllll} 0 & 3 & 6 & 9 & & & \\ 1 & 2 & 8 & 5 & 1 & 0 & 5 \\ 2 & 5 & 1 & 6 & & & \\ 3 & 8 & & & & & \\ 4 & 1 & & & & & \\ 5 & & & & & & \\ 6 & 2 & & & & \end{array} $$ a. Compute the sample mean, median, and mode for the data on distances driven. b. Compute the range, variance, and standard deviation for these data. c. Compute the first and third quartiles. d. Compute the interquartile range. Describe what properties the interquartile range has. When would the IQR be preferable to using the standard deviation when measuring variation?
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