EB/VSGO ARCS II PROGRAM Remedial Planning Activities at Selected Uncontrolled Hazardous Substance Disposal Sites Within EPA Region II (NY, NJ, PR, VI) FINAL SCREENING SITE INSPECTION (SSI) \ CAPTAIN'S COVE CONDOMINIUM SITE J GLEN COVE, NASSAU COUNTY NEW YORK SEPTEMBER 1995 VOLUME HI OF V EPA Contract 68-W8-0110 ERASCO An ENSERCH* Engineering and Construction Company 102923 EPA WORK ASSIGNMENT NO: 076-2JZZ EPA CONTRACT NO: 68-W8-0110 EBASCO SERVICES INCORPORATED ARCS 0 PROGRAM FINAL SCREENING SITE INSPECTION (SSI) CAPTAIN'S COVE CONDOMINIUM SITE GLEN COVE, NASSAU COUNTY NEW YORK SEPTEMBER 1995 VOLUME HI OF V NOTICE THE INFORMATION PROVIDED IN THIS DOCUMENT HAS BEEN FUNDED BY THE UNITED STATES ENVIRONMENTAL PROTECTION AGENCY (USEPA) UNDER ARCS E CONTRACT NO. 68-W8-0110 TO EBASCO SERVICES INCORPORATED (EBASCO). THIS DOCUMENT IS A DRAFT AND HAS NOT BEEN FORMALLY RELEASED BY EITHER EBASCO OR USEPA. AS A DRAFT, THIS DOCUMENT SHOULD NOT BE CITED OR QUOTED, AND IS BEING CIRCULATED FOR COMMENT ONLY. tech/preasm/ccove2.wp5 102924 REFERENCE NO. 26 102925 Probability and Statistics for Engineering and the Sciences SECOND EDITION Jay L. Devore California Polytechnic State University San Luis Obispo Brooks/Cole Publishing Company Monterey, California 102926 Contents Chapter 1 Introduction and Descriptive Statistics 1.1 An Overview of Probability and Statistics I 1.2 Pictorial and Tabular Methods in Descriptive Statistics 1.3 Measures of Location 12 1.4 Measures of Variability 20 Supplementary Exercises 29 Bibliography 30 Chapters Probability 31 2.1 Sample Spaces and Events 31 2.2 Axioms, Interpretations, and Properties of Probability 37 2.3 Counting Techniques 46 2.4 Conditional Probability 54 2.5 Independence 64 Supplementary Exercises 70 Bibliography 73 Chapter 3 Discrete Random Variables and Probability Distributions 74 3.1 Random Variables 74 3.2 Probability Distributions for Discrete Random Variables 79 3.3 Expected Values of Discrete Random Variables 88 3.4 The Binomial Probability Distribution 98 ix 102927 12 Chapter 1 Introduction and Descriptive Statistics appear as though a randomly selected Justice ». Using class intervals 0-<100, 100-<200, can be expected to serve 10 years or more? and so on, construct a relative frequency dis- Justify your answer. tribution for breaking strength. 10. An article in Environmental Concentration and b. If weaving specifications require a breaking Toxicology ("Trace Metals in Sea Scallops," vol. strength of at least 100 cycles, what propor- 19, pp. 326-1334) reported the amount of cad- tion of the yarn samples would be considered mium in sea scallops observed at a number of satisfactory? different stations in North Atlantic waters. The 13. Every score in the following batch of exam scores observed values follow: is in the 60's, 70's, 80's, or 90's. A stem and leaf 5.1, 14.4, 14.7, 10.8, 6.5, 5.7, 7:7, 14.1, 9.5, 3.7, display with only the four stems 6, 7, 8, and 9 8.9, 7.9, 7.9, 4.5, 10.1, 5.0, 9.6, 5.5, 5.1, 11.4, 8.0, would not give a very detailed description of the 12.1, 7.5, 8.5, 13:1, 6.4, 18.0, 27.0, 18.9, 10.8, distribution of scores. In such situations it is de- 13.1', 8.4, 16.9, 2.7, 9.6, 4.5, 12.4, 5:5, 12.7, 17.1 sirable to use repeated stems. Here we could re- peat the stem 6 twice, using 61 for scores in the a. Construct a frequency and relative frequency low 60's (leaves 0, 1, 2, 3, and 4) and 6h for distribution for the data set using 0.0-under scores in the high 60's (leaves 5, 6, 7, 8, and 9). 4.0 as the first class interval. Similarly, the other stems can be repeated twice b. Draw a histogram corresponding to the dis- (11, 7h, SI, &h, 91, and 9h) to obtain a display tributions of (a) so that area = relative consisting of eight rows. Construct such a display frequency. for the given scores. What feature of the data is 11. If in Example 1.5 the relative frequencies are highlighted by this display? computed to only two decimal places, do they add 74, 89, 80, 93, 64, 67, 72, 70, 66, 85, 89, 81, up to 1 ? 81, 71, 74, 82, 85, 63, 72, 81, 81, 95, 84, 81, 80, 12. In a study of warp breakage during the weav- 70, 69, 66, 60, 83, 85, 98, 84, 68, 90, 82, 69, 72, ing of fabric (Technometrics. 1982, p. 63), 100 87,88 pieces of yarn were tested. The number of cycles 14. For quantitative data, the cumulative frequency of strain to breakage was recorded for each yarn and cumulative relative frequency for a particu- sample. The resulting data is given below. lar class interval are the sum of frequencies and 86, 146, 251, 653, 98, 249, 400, 292, 131, 169, relative frequencies, respectively, for that interval 175, 176, 76, 264, 15, 364, 195, 262, 88, 264, and all intervals lying below it. If, for example, 157, 220, 42, 321, 180, 198, 38, 20, 61, 121, 282, there are four intervals with frequencies 9, 16, 224, 149, 180, 325, 250, 196, 90, 229, 166, 38, 13, and 12, then the cumulative frequencies are 337, 65, 151, 341, 40, 40, 135, 597, 246, 211, 9, 25, 38, and 50 and the cumulative relative fre- 180, 93, 315, 353, 571, 124, 279, 81, 186, 497, quencies are.. 18, .50, .76, and 1.00. Compute the 182, 423, 185, 229, 400, 338, 290, 398, 71, 246, cumulative frequencies and cumulative relative 185, 188, 568, 55, 55, 61, 244, 20, 284, 393, 396, frequencies for the data of Exercise 7. 203, 829, 239, 236, 286, 194, 277, 143, 198, 264, 105, 203, 124, 137, 135, 350, 193, 188 1.3 Measures of Location Having briefly studied tabular and pictorial methods for organizing and sum- marizing data, in this section and the next we will focus on numerical summary measures for a given data set. That is, from the data we try to extract several summarizing numbers, numbers that might serve to characterize the data set and convey some of its salient features. Our primary concern will be with numerical data, though some comments regarding categorical data appear at the end of the section. 102928 1.3 Measures of Location 13 Suppose, then, that our data set is of the form x,, x2,..,, XB, where each x, is a number. What features of such a set of numbers are of most interest and deserve emphasis? One important characteristic of a set of numbers is its loca- tion, and in particular its center. This section presents methods for describing the location of a data set, while in Section 1.4 we will turn to methods for measuring the variability in a set of numbers. The Mean For a given set of numbers xt, x2,..., xa, the most familiar and useful measure of the center is the mean, or arithmetic average of the set. Because we will almost always think of the x/s as constituting a sample, we will often refer to the arithmetic average as the sample mean and denote it by x. Definition The sample mean x of a set of numbers x,, x2,..., xn is given by •v _L v -l_ . _L v 2*•—> ' The value of x is in a sense more precise than the accuracy associated with any single observation. For this reason, we will customarily report the value of x using one digit of decimal accuracy beyond what is used in the individual x,'s. Example 1.7 The amount of light reflectance by leaves has been used for various purposes, including evaluation of turf color, estimation of nitrogen status, and measure- ment of biomass. The paper "Leaf Reflectance-Nitrogen-Chlorophyll Relations in Buffelgrass" (Photogrammetric Engineering and Remote Sensing. 1985, pp. 463-466) gave the following observations, obtained using spectrophotogram- metry, on leaf reflectance under specified experimental conditions: 15.2, 16.8, 12.6, 13.2, 12.8, 13.8, 16.3, 13.0, I2.7, 15.8, 19.2, 12.7, 15.6, 13.5, 12.9 The sum of these 15 x,'s is Sx, = 15.2 + 16.8 + • • • + 12.9 = 216.1, so the value of the sample mean is 15 216.1 = 14.41 15 15 A physical interpretation of x will demonstrate how it measures the loca- tion (center) of a sample. Think of drawing and scaling a horizontal measure- ment axis, and then represent each sample observation by a one-pound weight placed at the corresponding point on the axis. The only point at which a ful- crum can be placed to balance the system of weights is the point corresponding to the value of x (see Figure 1.6). 14 Chapter 1 Introduction and Descriptive Statistics x = 14.41 12 13 14 15 16 17 18 19 Figure 1.6 The mean as the balance point for a system of weights Just as x represents the average value of the observations in a sample, the average of all values in the population can be calculated. This average is called the population mean and is denoted by the Greek letter fi. When there are N values in the population (a finite population), then fi — (sum of the N popula- tion values)/7V. Example 1.8 Exercise 9 listed the length of service of each U. S. Supreme Court Justice whose service terminated before 1978.