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Bài giảng thống kê ứng dụng trong quản lý xây dựng Lê Hoài Long 11Population and sample 12Variables 13Measures of data 14Pattern of data 15Scales of measurement 16Between variables 21Data collection 22Probability 23Variable 24Sampling distribution 31One sample estimation 32One sample hypothesis testing 33Two sample 34Multisample estimation and testing 35Nonparametric techniques
VARIABLE AND DISTRIBUTION Part – Section Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn When the numerical value of a variable is determined by a chance event, that variable is called a random variable Random variables can be discrete or continuous Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn A probability distribution is a table or an equation that links each possible value that a random variable can assume with its probability of occurrence Two types: Discrete Probability Distributions Continuous Probability Distributions Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Discrete Probability Distributions The probability distribution of a discrete random variable can always be represented by a table Given a probability distribution, you can find cumulative probabilities Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Continuous Probability Distributions The probability distribution of a continuous random variable is represented by an equation, called the probability density function (pdf) All probability density functions satisfy the following conditions: The random variable Y is a function of X; that is, y = f(x) The value of y is greater than or equal to zero for all values of x The total area under the curve of the function is equal to one Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Continuous Probability Distributions The probability that a continuous random variable falls in the interval between a and b is equal to the area under the pdf curve between a and b There are an infinite number of values between any two data points As a result, the probability that a continuous random variable will assume a particular value is always zero Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Mean of a Discrete Random Variable The mean of the discrete random variable X is also called the expected value of X, denoted by E(X): E ( X ) x xi P( xi ) where xi is the value of the random variable for outcome i, μx is the mean of random variable X, and P(xi) is the probability that the random variable will be outcome i Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Median of a Discrete Random Variable The median of a discrete random variable is the "middle" value It is the value of X for which P(X < x) is greater than or equal to 0.5 and P(X > x) is greater than or equal to 0.5 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Example The number of hits made by each player is described by the following probability distribution Number of hits, x Probability, P(x) 0.10 0.20 0.30 0.25 0.15 What is the mean of the probability distribution? What is the median? Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn Variability of a Discrete Random Variable The equation for computing the variance of a discrete random variable is: xi E ( X ) P( xi ) Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 10 Like the standard normal distribution, the t distribution is symmetric and we also deal with one and two-tailed probabilities Use t statistic instead of z Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 92 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 93 We select a random sample of size n from a normal population, having a standard deviation equal to σ We find that the standard deviation in our sample is equal to s Given these data, we can define a statistic, called chi-square: (n 1) s 2 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 94 If we repeated this experiment an infinite number of times, we could obtain a sampling distribution for the chi-square statistic 1 2 f ( ) fo e where Χ2 is the chi-square statistic, v = n - is the number of degrees of freedom f0 is defined, so that the area under the chisquare curve is equal to one Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 95 The mean of the distribution is equal to the number of degrees of freedom: μ = v The variance is equal to two times the number of degrees of freedom: σ2 = v When the degrees of freedom are greater than or equal to 2, the maximum value for f occurs when Χ2 = v - As the degrees of freedom increase, the chisquare curve approaches a normal distribution Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 96 The area under the curve between and a particular chisquare value is a cumulative probability associated with that chi-square value For example, in the figure, the shaded area represents a cumulative probability associated with a chi-square statistic equal to A; that is, it is the probability that the value of a chi-square statistic will fall between and A Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 97 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 98 Given two independent chi-square random variables, then the ratio of these variables with each divided by its respective degrees of freedom is a continuous F variable, that has an F distribution with two forms of degrees of freedom 2 1 F 2 2 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 99 The probability function for the F distribution is: 1 f ( F ) cF 1 1F 1 2 2 Where c is a constant that is a function of v1 and v2 that is needed to make the total area under the F distribution equal to Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 100 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 101 CHEBYSHEV’S THEOREM Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 102 CHEBYSHEV’S THEOREM For any number k 1, and a set of data x1, x2, …, xn, the proportion of the measurements that lies within k standard deviations of their mean will be at least: 1 k Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 103 Frequency CHEBYSHEV’S THEOREM 1 k -k Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn +k 104 CHEBYSHEV’S THEOREM Example: A manufacturer wants to claim that at least 96% of their products last from 95 days to 105 days If they test a sample of 1000 products and get a mean ofx = 100 days, then what is the maximum value possible for the sample standard deviation s if they are to make the 96% claim? Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 105 CHEBYSHEV’S THEOREM Please justify the empirical rule of normal distribution using the theorem Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 106 ... Hypergeometric distribution Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 14 BINOMIAL DISTRIBUTION Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 15 Binomial Experiment The... ( x) n C x p (1 p ) xa n x xa Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 21 Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 22 Example: Flooding of a road Suppose... requirements The producer' s risk Pr Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 28 NEGATIVE BINOMIAL Lecturer: Le Hoai Long (Ph.D.) lehoailong@hcmut.edu.vn 29 Negative Binomial Experiment