Chapter 01

# Chapter 01 - Chapter 1 Data and Distributions Section 1.2...

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Chapter 1 Data and Distributions Section 1.2 1. (a) Minitab generates the following stem-and-leaf display of this data: Stem-and-leaf of C1 N = 27 Leaf Unit = 0.10 1 5 9 6 6 33588 (11) 7 00234677889 10 8 127 7 9 077 4 10 7 3 11 368 The left most column in the Minitab printout shows the cumulative numbers of observations from each stem to the nearest tail of the data. For example, the 6 in the second row indicates that there are a total of 6 data points contained in stems 6 and 5. Minitab uses parentheses around 11 in row three to indicate that the median (described in Chapter 2, Section 2.1) of the data is contained in this stem. A value close to 8 is representative of this data. What constitutes large or small variation usually depends on the application at hand, but an often-used rule of thumb is: the variation tends to be large whenever the spread of the data (the difference between the largest and smallest observations) is large compared to a representative value. Here, 'large' means that the percentage is closer to 100% than it is to 0%. For this data, the spread is 11 - 5 = 6, which constitutes 6/8 = .75, or, 75%, of the typical data value of 8. Most researchers would call this a large amount of variation. (b) The data display is not perfectly symmetric around some middle/representative value. There tends to be some positive skewness in this data. (c) In Chapter 1, outliers are data points that appear to be very different from the pack. Looking at the stem-and-leaf display in part (a), there appear to be no outliers in this data. (Chapter 2 gives a more precise definition of what constitutes an outlier). 1 - 1

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Chapter 1 (d) From the stem-and-leaf display in part (a), there are 3 leaves associated with the stem of 11, which represent the 3 data values that greater than or equal to 11 (and hence, strictly greater than 10). Therefore, the proportion of data values that exceed 10 is 3/27 = .148, or, about 15%. 2. (a) Using the same stem and leaf units as in Exercise 1, a comparative stem-and- leaf display of this data is: data from Exercise 1 (n=27) data from Exercise 2 (n=20) 9 5 8 33588 6 16 00234677889 7 012488 127 8 13359 077 9 278 7 10 368 11 2 12 6 13 14 1 From this display, the cylinder data appears to be even more positively skewed than the data from Exercise 1. The data value 14.1 appears to be an outlier. From the set-and-leaf display, there are 3 values in the cylinder data that have stems of 11 or larger, so there the proportion of cylinder strengths that exceed 10 is 3/20 = .15, or, 15%. (b) Both data sets have approximately the same representative value of about 8 MPa and both stem-and-leaf displays exhibit positive skewness. The spread of the cylinder data is larger than that of the beam data and the cylinder data also appears to contain an outlier. 1 - 2
Chapter 1 3. A Minitab stem-and-leaf display of this data is: Stem-and-leaf of C1 N = 36 Leaf Unit = .01 1 3 1 6 3 56678 18 4 000112222234 18 4 5667888 11 5 144 8 5 58 6 6 2 5 6 6678 1 7 1 7 5 Another method of denoting the pairs of stems having equal values is to denote the first stem by L, for 'low', and the second stem by H, for 'high'. Using this notation, the stem-and-leaf display would appear as follows: 3L

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