1.4.5 - Working with Box-and-Whisker Plots

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Maharishi University of Management *

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Statistics

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Apr 3, 2024

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Working with Box-and-Whisker Plots 1. The data below shows the number of raisins in each of 14 boxes (1/2 oz.) of three different brands of raisins: (8 points) Sun Tyrant: 25, 28, 25, 28, 29, 24, 28, 24, 24, 28, 30, 24, 22, 27 Sun Acrid: 29, 31, 29, 26, 28, 22, 25, 29, 29, 27, 28, 23, 26, 29 Laminatekist: 25, 26, 26, 26, 26, 28, 27, 26, 25, 28, 24, 28, 27, 25 A. Compute the median, lower and upper quartiles, and interquartile range for each data set. NOTE: We are using the following definitions in this Assignment and course: when there's an odd number of observations in a group, the median is the middle number. When there's an even number of observations in a group, the median is (n+1)/2. The Interquartile Range, IQR, is Q3 – Q1, where Q3 is the median number of the top 50% and Q1 is the median of the bottom 50% of scores. (We are not using fences.) Five-number summary for the following: Sun Tyrant lower quartile: 24 median: 26 upper quartile: 28 IQR: 4 Sun Acrid lower quartile: 26 median: 28 upper quartile: 29 IQR: 3 Laminatekist lower quartile: 25 median: 26 upper quartile: 27 IQR: 2 B. Construct three different modified box-and-whisker plots and place them on the same horizontal axis, labeling the axes, and labeling Q1, Q3, the median and the mean on the boxes.

C. What conclusions can you make about the shape of the distributions by looking at the three box plots? Are the distributions symmetric or skewed in one direction? Justify your answer. The first boxplot is roughly symmetric; second is skewed left; third is roughly symmetric. Also, the means for the first and third boxplots are closer to their medians compared to the second one. Thus, Sun Acrid has a distribution of number of raisins near the upper end compared to the other two brands. D. For which brand are you more likely to end up with a box with a fewer raisins? Justify your answer. Sun Tyrant; it has a roughly symmetrical distribution, and its lower quartile is the lowest compared to the other two brands, thus have more likeliness to end up with a box with fewer raisins. 2. Here is a set of data of college GPAs from a sample of students who completed an AP course in statistics in their senior year at a fictional high school: (9 points) 3.8, 3.8, 3.9, 3.9, 3.5, 3.5, 3.0, 3.5, 3.8, 4.0, 3.5, 2.5, 2.4, 1.9, 4.0, 3.8, 3.7, 3.5, 3.9, 3.2 A. Construct a histogram of the GPA data using a class width of 0.3, and a minimum x -value of 1.9, and be sure to label both axes. B. Construct a modified box-and-whisker plot using the GPA data and report the values of the five-number summary, labeling the axes, and labeling Q1, Q3, the median and the mean on the boxes.

C. Describe the shape of the distribution of GPAs in terms of symmetry (is it skewed?) and outliers. Use evidence from the two types of plots to support your answer. Left skewed as shown in the histogram; outliers can be seen plotted on the left side in the boxplot. 3. The Houses of Parliament in Ghyronmia (a hypothetical small country) have two political parties: the Purple Party and the Chartreuse Party. Twenty different “tax cut” bills came up in the most recent Parliamentary session. If senators voted yes , they were in support of a tax cut. The number of times (out of a possible 20) each senator voted yes is shown below: (8 points) Number of times senators from each party voted yes to a tax cut bill: Purple : 17, 18, 20, 15, 12, 15, 17, 19, 20, 17, 20, 18, 15, 9, 14, 16, 17, 14, 3, 15 Chartreuse : 11, 13, 12, 9, 17, 11, 10, 14, 7, 8, 8, 11, 12, 12, 13, 12, 14, 10, 10, 10 A. Construct modified box-and-whisker plots for each data set on the same horizontal axis. Please indicate the values of Q1, median, and Q3 on the different plots. B. Compute the IQR for each data set. Purple IQR: 3.5

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Chartreuse IQR: 2.5 C. Compare the shapes of the two distributions (skew and outliers) and the amount of variation in the distributions. What do these plots tell you about the policies of Chartreuse and Purples in this particular state during this particular legislative session? What do the outliers tell you about the behavior of a few of the senators? The purple party has more outliers and more variation (since the IQR is higher) than the chartreuse party. The plots tell me that the purple party was more in support of the tax cut bill compared to the chartreuse party at that time. Some senators in the purple party did not want some of the tax cut bills to be passed as. A member of the chartreuse party wanted overall more bills to be passed than the rest of his party.