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Chapter 3: Problem 104

The following data give the numbers of computer keyboards assembled at the Twentieth Century Electronics Company for a sample of 25 days. \(\begin{array}{llllllllll}45 & 52 & 48 & 41 & 56 & 46 & 44 & 42 & 48 & 53 \\\ 51 & 53 & 51 & 48 & 46 & 43 & 52 & 50 & 54 & 47 \\ 44 & 47 & 50 & 49 & 52 & & & & & \end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

Short Answer

Expert verified
After drawing the box-and-whisker plot, examine its features. If the median line within the 'box' is closer to the upper quartile (Q3) it could mean the data is skewed to the left (negative skewness). If the median line is closer to the lower quartile (Q1), the data might be skewed to the right (positive skewness). However, this would only be an estimate as we do not have the mean of the data to draw a definitive conclusion about skewness.

Step by step solution

01

Sorting the Data

First, to construct a box plot, arrange the given data points in ascending order.
02

Finding the Five Number Summary

Calculate the minimum, Q1 (the 25th percentile), the median (Q2 or the 50th percentile), Q3 (the 75th percentile), and the maximum. Getting the median of whole data and then getting median of lower half and upper half separately helps find Q1 and Q3, respectively.
03

Creating the Box Plot

Draw a number line from minimum to maximum. Mark off the Median, the Q1, and the Q3 on the line and create a box around the quartiles. The whiskers are added by stretching out from the box and up to the maximum and minimum.
04

Commenting on Skewness

Examine the box plot. If the median (Q2) line is closer to Q3 or to Q1, then the data might be right-skewed or left-skewed, respectively. This provides a crude indication of skewness that is not as precise as comparing mean to median, but can be used in this condition where mean isn't provided.

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Most popular questions from this chapter

Refer to Exercise \(2.94\), which listed the alcohol content by volume for each of the 13 varieties of beer produced by Sierra Nevada Brewery. Those data are reproduced here: \(\begin{array}{llllllllll}4.4 & 5.0 & 5.0 & 5.6 & 5.6 & 5.8 & 5.9 & 5.9 & 6.7 & 6.8\end{array}\) \(\begin{array}{lll}6.9 & 7.0 & 9.6\end{array}\) Calculate the range, variance, and standard deviation.

The following data give the numbers of new cars sold at a dealership during a 20-day period. \(\begin{array}{rrrrrrrrrr}8 & 5 & 12 & 3 & 9 & 10 & 6 & 12 & 8 & 8 \\ 4 & 16 & 10 & 11 & 7 & 7 & 3 & 5 & 9 & 11\end{array}\) Make a box-and-whisker plot. Comment on the skewness of these data.

The following data give the hourly wage rates of eight employees of a company. \(\begin{array}{lllllll}\$ 22 & 22 & 22 & 22 & 22 & 22 & 22\end{array}\) 22 Calculate the standard deviation. Is its value zero? If yes, why?

A sample of 3000 observations has a bell-shaped distribution with a mean of 82 and a standard deviation of \(16 .\) Using the empirical rule, find what percentage of the observations fall in the intervals \(\bar{x} \pm 1 s, \bar{x} \pm 2 s\), and \(\bar{x} \pm 3 s .\)

The range, as a measure of spread, has the disadvantage of being influenced by outliers. Illustrate this with an example.
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