ANALYSIS FOR YEARS 1911 1969

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For these years, we nd the least squares regression line to be Speed = 231589 + 12544 Year and r 2 = 0980022, a remarkable t For the median median line, we use the points (19135, 8060) , (1930, 10045), and (1958, 135601), the last point being the median point for the years 1947 through 2008 We nd the median median line to be y = 123598x 228463 The lines are very closely parallel, but have slightly different intercepts Predictions based upon them will be very close These lines are shown in Figure 1314

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13

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Least Squares, Medians, and the Indy 500

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200 180 Speed 160 140 120 100 1940 1960 Year 1980 2000

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CONCLUSIONS

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The data from the winning speeds at the Indianapolis 500-mile race provide a fairly realistic exercise when one is confronted with a genuine set of data Things rarely work out as well as they do with textbook cases of arranged or altered data sets We nd that in this case, the median median line is a ne approximation of the data for the early years of the data and that the least squares line is also ne for the early years of the data; neither is acceptable for the later years when we speculate that alterations in the aerodynamics of the cars and time spent under the yellow ag produce speeds that vary considerably from a straight line prediction

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EXPLORATIONS

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1 Using the three median points for the Indy 500 data, show that method 3 is a valid procedure for nding the median median line 2 Using the three median points for the Indy 500 data, nd the least squares line for these points 3 Find the analysis of variance partition for the least squares line in Exploration 2 4 Analyze the Indy 500 data for 1998 2008 by nding both the median median and the least squares lines Show the partitions of the total sum of squares, SST, in each case

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Sampling

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CHAPTER OBJECTIVES:

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r r r r r r

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to show some properties of simple random sampling to introduce strati ed sampling to nd some properties of strati ed sampling to see how proportional allocation works to discuss optimal allocation to nd some properties of proportional and optimal allocation

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One of the primary reasons that statistics has become of great importance in science and engineering is the knowledge we now have concerning sampling and the conclusions that can be drawn from samples It is perhaps a curious and counterintuitive fact that knowledge about a population or group can be found with great accuracy by examining a sample only part of the population or group Almost all introductory courses in statistics discuss only simple random sampling In simple random sampling every item in the population is given an equal chance of occurring in the sample, so every item in the population is treated exactly equally It may come as a surprise to learn that simple random sampling can often be improved upon; that is, other sampling procedures may well be more ef cient in providing information about the population from which the sample is selected In these procedures, not all the sampled items are treated equally! In addition to simple random sampling, we will discuss strati ed sampling and both proportional allocation and optimal allocation within strati ed sampling We start with a very small example so that ideas become clear

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A Probability and Statistics Companion, John J Kinney Copyright 2009 by John Wiley & Sons, Inc

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Sampling

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EXAMPLE 141

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High and Middle Schools

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An urban school district is interested in discovering some characteristics of some of its high and middle schools We emphasize from the beginning that we assume we know the entire population In practice, however, we would never know this (or else sampling is at best an idle exercise) The details of the population are given in Table 141 Table 141 School enrollment 1667 2002 1493 1802 1535 731 834 699 School type High High High High High Middle Middle Middle

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The mean enrollment is x = 1345375 and the standard deviation is x = 4818737 We want to show how these statistics can be estimated by taking a sample of the schools The subscript x is used to distinguish the population from the samples we will select We rst consider simple random sampling