Advanced Engineering Mathematics

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Unformatted text preview: Part G. PROBABILITY, STATISTICS CHAPTER 24 Data Analysis. Probability Theory SECTION 24.1. Data Representation. Average. Spread, page 993 Purpose. To discuss standard graphical representations of data in statistics. To introduce concepts that characterize the average size of the data values and their spread (their variability). Main Content, Important Concepts Stem-and-leaf plot Histogram Boxplot Absolute frequency, relative frequency Cumulative relative frequency Outliers Mean Variance, standard deviation Median, quartiles, interquartile range Comment on Content The graphical representations of data to be discussed in this section have become standard in connection with statistical methods. Average size and variability give the two most important general characterizations of data. Relative frequency will motivate probability as its theoretical counterpart. This is a main reason for presenting this material here before the beginning of our discussion of probability in this chapter. Randomness is not mentioned in this section because the introduction of samples (random samples) as a concept can wait until Chap. 25 when we shall need them in connection with statistical methods. The connection with this section will then be immediate and will provide no difficulty or duplication. SOLUTIONS TO PROBLEM SET 24.1, page 996 2. q L 5 2, q M 5 5, q U 5 6 4. q L 5 10.0, q M 5 11.6, q U 5 12.4 6. q L 5 2 0.52, q M 5 2 0.19, q U 5 0.24 8. q L 5 85, q M 5 87, q U 5 89 10. q L 5 q M 5 14, q U 5 14.5 12. x w 5 4.3, s 5 2.541, IQR 5 4 14. x w 5 2 0.064, s 5 0.542, IQR 5 0.76 16. x w 5 12.6 but q M 5 7. The data are not sufficiently symmetric. s 5 9.07, IQR 5 17 18. x min % x j % x max . Now sum over j from 1 to n . Then divide by n to get x min % x w 5 x max . 20. Points to consider are the amounts of calculation, the size of the data (in using quartiles we lose information—the larger the number of data points, the more more information we lose), and the symmetry and asymmetry of the data. In the case of symmetry we 374 im24.qxd 9/21/05 5:14 PM Page 374 have better agreement between quartiles on the one hand and mean and variance on the other, as in the case of data with considerable deviation from symmetry. SECTION 24.2. Experiments, Outcomes, Events, page 997 Purpose. To introduce basic concepts needed throughout Chaps. 24 and 25. Main Content, Important Concepts Experiment Sample space S , outcomes, events Union, intersection, complements of events Mutually exclusive events Representation of sets by Venn diagrams Comment on Content To make the chapter self-contained, we explain the modest amount of set-theoretical concepts needed in the next sections, although most students will be familiar with these matters....
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