Probability, Statistics, and Reliability for Engineers and Scientists FUNDAMENTALS OF STATISTICAL ANALYSIS
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1 CHAPTER Probability, Statistics, and Reliability for Engineers and Scientists FUNDAMENTALS OF STATISTICAL ANALYSIS Second Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering 8a Probability and Statistics for Civil Engineers Department of Civil and Environmental Engineering University of Maryland, College Park CHAPMAN HALL/CRC CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 1 Definition Statistics: is the estimation of certain parameters (i.e., mean,cov, distribution type) needed to quantify uncertainty and to describe the probability functions. 1
2 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 2 Estimation of Parameters of a Distribution Once the distribution type of a random variable is assumed, it is necessary to define it uniquely by evaluating its parameters. Some distributions have only one parameter, while others have multiple. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 3 Examples: Distribution Parameters Normal Distribution: Two parameters, µ X and σ X Lognormal Two parameters, µ Y and σ Y (or λ and ζ) Poisson One parameter, λ Beta Four parameters, a, b, q, and r 2
3 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 4 Sampling Values of random variables obtained from sample measurements are used to estimate the parameters of a distribution. They are also used to in making important engineering decision. For example, in determining the maximum wind speed for the design of a tall building, past records of measured wind velocities near the building site are important. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 5 Sampling Samples of river water are collected to estimate the average level of a pollutant in the entire river at that location. The average of sample measurements of the compressive strength of concrete collected during the pouring of a large concrete slab, such as a deck of of a parking garage, is used to help decide whether or not the deck strength met the specifications. 3
4 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 6 Sampling The estimated mean for a random variable is considered by itself to be a random variable, because different samples about the random variable can produce different estimated mean values. Hence, randomness in the estimated mean. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 7 Sampling When a sample of n measurements of a random variable is collected, the n values are not necessarily identical. The sample is characterized by variation. If two different samples of 5 measurements of concrete strength are collected, their mean values would not necessarily be identical. 4
5 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 8 Mean StDev Sample Sample CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 9 Assume that the building code requires a mean compressive strength of 3500 psi. Since the mean of 3442 psi of Sample1 is less than 3500 psi Should we conclude that the garage deck dose not meet the specifications? 5
6 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 10 Unfortunately, decision making is not that simple. If a third sample of 5 measurements had been randomly collected from other locations on the garage deck, the following values as just likely to have been obtained: 3720, 3440, 3590, 3270, and 3610 psi. This sample would have different mean and different standard deviation as shown next. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 11 Mean StDev Sample Sample Sample
7 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 12 The third sample (Sample 3) produces a mean of 3526 psi and standard deviation of psi. In this case, the mean value is greater than the specified value. The question now arises: Can we conclude that the concrete is of an adequate strength? Unfortunately, we cannot conclude with certainty that the strength is adequate. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 13 The fact that different samples lead to different means is an indication that we cannot conclude that the design specification is not met just because the sample mean is less than the design standard. 7
8 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 14 A systematic decision process is needed to take into account the variation that can be expected from one sample to another. The decision process must also capable to reflect the risk of making incorrect decision. This decision making can be made using hypothesis testing. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 15 Sample Parameters 1 X = n S = n i= 1 x 1 n 1 i n ( xi X ) i= 1 2 Population Parameters For normal distribution : µ σ X X 8
9 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 16 The data that are collected represent sample information but it is not complete by itself. Predictions are not made directly from the sample. The intermediate step between sampling and prediction is the identification of the underlying population. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 17 The sample is used to identify the population. Then, the population is used to make prediction or decision. The sample-to-population-to-predictions sequence is true for both univariate, bivariate, and multivariate methods. 9
10 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 18 Population Models A known model or function is often used to represent the population. The normal and logmormal distributions are commonly used to model the population for a univariate problem For bivariate and multivariate prediction, linear and power model are assumed. CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 19 Example: Population Models Linear Model Power Model Y ˆ = a + bx Y ˆ = ax b 10
11 CHAPTER 8a. FUNDAMENTALS OF STATISTICAL ANALYSIS Slide No. 20 When using a probability function to represent the population, it is necessary to estimate the parameters Normal Distribution Estimate : µ and σ Lognormal Estimate : λ and ζ Exponential Estimate : λ 11
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