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DISPLAYING AND SUMMARIZING DATA
98 5 6 39 31
46 129 17 1 64
40 121 88 102 50
123 50 20 37 65
75 191 110 28 44
47 6 43 60 12
150 16 182 32 5
106 32 26 87 137
44 13 18 69 107
5 53 54 173 118
(a) Form a stem-and- leaf diagram of the measurements.
(b) Find the median, lower quartile, and upper quartile of the measurements.
(c) Sketch a boxplot of the measurements.
(d) Put the measurements in a frequency table with the following classes:
0 < x < 20
20 < x < 40
40 < x < 60
60 < x < 80
80 < x < 100
100 < x < 200
(e) Construct a histogram of the measurements.
(f) Construct a cumulative frequency polygon of the measurements. Show the median and quartiles.
3.5 A random sample of 50 families reported the dollar amount they had available as a liquid cash reserve. The data have been put in the following frequency table:
0 <x< 500 17
500 < x < 1000 15
1000 < x < 2000 7
2000 < x < 4000 5
4000 < x < 6000 3
6000 < x < 10000 3
(a) Construct a histogram of the measurements.
(b) Construct a cumulative frequency polygon of the measurements. Show the median and quartiles.
(c) Calculate the grouped mean for the data.
3.6 In this exercise we see how the default setting in the Minitab boxplot command can be misleading, since it doesn’t take the sample size into account. We will generate three samples of different sizes from the same distribution, and compare their Minitab boxplots. Generate 250 normal (0,1) observations and put them in column c1 by pulling down the calc menu to the random data command over to normal and filling in the dialog box. Generate 1000 normal (0,1) observations the same way and put them in column c2, and generate 4000 normal (0,1) observations the same way and put them in column c3. Stack these three columns by pulling down the manip menu down to stack/unstack and over to stack columns and filling in the dialog box to put the stacked column into c4, with subscripts into c5. Form stacked boxplots by pulling down graph menu to boxplot command and filling in dialog box. Y is c4 and x is c5.
(a) What do you notice from the resulting boxplot?
(b) Which sample seems to have a heavier tail?
(c) Why is this misleading?
(d) Redo the boxplot highlighting the outlier symbol in the dialog box, and clicking on edit attributes and select dot.
(e) Is the graph still as misleading as the original?
3.7 McGhie and Barker (1984) collected 100 slugs from the species Limax maximus around Hamilton, New Zealand. They were preserved in a relaxed state, and their length in mm and weight in gm were recorded. Thirty of the observations are shown below. The full data are in the Minitab worksheet slug.mtw.
54 DISPLAYING AND SUMMARIZING DATA
length weight length weight length weight
(mm) (gm) (mm) (gm) (mm) (gm)
73 3.68 21 0.14 75 4.94
78 5.48 26 0.35 78 5.48
75 4.94 26 0.29 22 0.36
69 3.47 36 0.88 61 3.16
60 3.26 16 0.12 59 1.91
74 4.36 35 0.66 78 8.44
85 6.44 36 0.62 90 13.62
86 8.37 22 0.17 93 8.70
82 6.40 24 0.25 71 4.39
85 8.23 42 2.28 94 8.23
(a) Plot weight on length using Minitab. What do you notice about the shape of the relationship?
(b) Often when we have a nonlinear relationship, we can transform the variables by taking logarithms and achieve linearity. In this case, weight is related to volume which is related to length times width times height. Taking logarithms of weight and length should give a more linear relationship. Plot log(weight) on log(length) using Minitab. Does this relationship appear to be linear?
(c) From the scatterplot of log(weight) on log(length) can you identify any points that do not appear to fit the pattern?
Logic, Probability, and Uncertainty
Most situations we deal with in everyday life are not completely predictable. If I think about the weather tomorrow at noon, I cannot be certain whether it will or will not be raining. I could contact the Meteorological Service and get the most up to date weather forecast possible, which is based on the latest available data from ground stations and satellite images. The forecast could be that it will be a fine day. I decide to take that forecast into account, and not take my umbrella. Despite the forecast it could rain and I could get soaked going to lunch. There is always uncertainty.
In this chapter we will see that deductive logic can only deal with certainty. This is of very limited use in most real situations. We need to develop inductive logic that allows us to deal with uncertainty.
Since we can’t completely eliminate uncertainty, we need to model it. In real life when we are faced with uncertainty, we use plausible reasoning. We adjust our belief about something, based on the occurrence or nonoccurrence of something else. We will see how plausible reasoning should be based on the rules of probability which were originally derived to analyze the outcome of games based on random chance. Thus the rules of probability extend logic to include plausible reasoning where there is uncertainty.