Applied Statistics I

Transcription

1 Applied Statistics I Liang Zhang Department of Mathematics, University of Utah July 8, 2008 Liang Zhang (UofU) Applied Statistics I July 8, / 15

2 Distribution for Sample Mean Liang Zhang (UofU) Applied Statistics I July 8, / 15

3 Distribution for Sample Mean Example (Problem 54) Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.65 and standard deviation.85 (suggested in Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants, Water Research, 1984: ). Liang Zhang (UofU) Applied Statistics I July 8, / 15

4 Distribution for Sample Mean Example (Problem 54) Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.65 and standard deviation.85 (suggested in Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants, Water Research, 1984: ). a. If a random sample of 25 specimens is selected, what is the probability that the sample average sediment density is at most 3.00? Liang Zhang (UofU) Applied Statistics I July 8, / 15

5 Distribution for Sample Mean Example (Problem 54) Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.65 and standard deviation.85 (suggested in Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants, Water Research, 1984: ). a. If a random sample of 25 specimens is selected, what is the probability that the sample average sediment density is at most 3.00? b. How large a sample size would be required to ensure that the above probability is at least.99? Liang Zhang (UofU) Applied Statistics I July 8, / 15

6 Distribution for Sample Mean Liang Zhang (UofU) Applied Statistics I July 8, / 15

7 Distribution for Sample Mean The Central Limit Theorem (CLT) Let X 1, X 2,..., X n be a random sample from a distribution with mean value µ and standard deviation σ. Then if n is sufficiently large, X has approximately a normal distribution with mean value µ and standard deviation σ/ n, and T 0 also has approximately a normal distribution with mean value nµ and standard deviation nσ. The larger the value of n, the better the approximation. Liang Zhang (UofU) Applied Statistics I July 8, / 15

8 Distribution for Sample Mean The Central Limit Theorem (CLT) Let X 1, X 2,..., X n be a random sample from a distribution with mean value µ and standard deviation σ. Then if n is sufficiently large, X has approximately a normal distribution with mean value µ and standard deviation σ/ n, and T 0 also has approximately a normal distribution with mean value nµ and standard deviation nσ. The larger the value of n, the better the approximation. Remark: 1. As long as n is sufficiently large, CLT is applicable no matter X i s are discrete random variables or continuous random variables. Liang Zhang (UofU) Applied Statistics I July 8, / 15

9 Distribution for Sample Mean The Central Limit Theorem (CLT) Let X 1, X 2,..., X n be a random sample from a distribution with mean value µ and standard deviation σ. Then if n is sufficiently large, X has approximately a normal distribution with mean value µ and standard deviation σ/ n, and T 0 also has approximately a normal distribution with mean value nµ and standard deviation nσ. The larger the value of n, the better the approximation. Remark: 1. As long as n is sufficiently large, CLT is applicable no matter X i s are discrete random variables or continuous random variables. 2. How large should n be such that CLT is applicable? Liang Zhang (UofU) Applied Statistics I July 8, / 15

10 Distribution for Sample Mean The Central Limit Theorem (CLT) Let X 1, X 2,..., X n be a random sample from a distribution with mean value µ and standard deviation σ. Then if n is sufficiently large, X has approximately a normal distribution with mean value µ and standard deviation σ/ n, and T 0 also has approximately a normal distribution with mean value nµ and standard deviation nσ. The larger the value of n, the better the approximation. Remark: 1. As long as n is sufficiently large, CLT is applicable no matter X i s are discrete random variables or continuous random variables. 2. How large should n be such that CLT is applicable? Generally, if n > 30, CLT can be used. Liang Zhang (UofU) Applied Statistics I July 8, / 15

11 Distribution for Sample Mean Liang Zhang (UofU) Applied Statistics I July 8, / 15

12 Distribution for Sample Mean Example (Problem 49) There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 6 min and a standard deviation of 6 min. Liang Zhang (UofU) Applied Statistics I July 8, / 15

13 Distribution for Sample Mean Example (Problem 49) There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 6 min and a standard deviation of 6 min. a. If grading times are independent and the instructor begins grading at 6:50pm and grades continuously, what is the (approximate) probability that he is through grading before the 11:00pm TV news begins? Liang Zhang (UofU) Applied Statistics I July 8, / 15

14 Distribution for Sample Mean Example (Problem 49) There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 6 min and a standard deviation of 6 min. a. If grading times are independent and the instructor begins grading at 6:50pm and grades continuously, what is the (approximate) probability that he is through grading before the 11:00pm TV news begins? b. If the sports report begins at 11:10pm, what is the probability that he misses part of the report if he waits unitl grading is done before turning on the TV? Liang Zhang (UofU) Applied Statistics I July 8, / 15

15 Distribution for Sample Mean Liang Zhang (UofU) Applied Statistics I July 8, / 15

16 Distribution for Sample Mean The original version of CLT The Central Limit Theorem (CLT) Let X 1, X 2,... be a sequence of i.i.d. random variables from a distribution with mean value µ and standard deviation σ. Define random variables n i=1 Y n = X i nµ for n = 1, 2,... nσ Then as n, Y n has approximately a normal distribution. Liang Zhang (UofU) Applied Statistics I July 8, / 15

17 Distribution for Sample Mean Liang Zhang (UofU) Applied Statistics I July 8, / 15

18 Distribution for Sample Mean Corollary Let X 1, X 2,..., X n be a random sample from a distribution for which only positive values are possible [P(X i > 0) = 1]. Then if n is sufficiently large, the product Y = X 1 X 2 X n has approximately a lognormal distribution. Liang Zhang (UofU) Applied Statistics I July 8, / 15

19 Distribution for Linear Combinations Liang Zhang (UofU) Applied Statistics I July 8, / 15

20 Distribution for Linear Combinations Proposition Let X 1, X 2,..., X n have mean values µ 1, µ 2,..., µ n, respectively, and variances σ 2 1, σ2 2,..., σ2 n, respectively. 1.Whether or not the X i s are independent, E(a 1 X 1 + a 2 X a n X n ) = a 1 E(X 1 ) + a 2 E(X 2 ) + + a n E(X n ) 2. If X 1, X 2,..., X n are independent, = a 1 µ 1 + a 2 µ a n µ n V (a 1 X 1 + a 2 X a n X n ) = a 2 1V (X 1 ) + a 2 2V (X 2 ) + + a 2 nv (X n ) = a 1 σ a 2 σ a n σ 2 n Liang Zhang (UofU) Applied Statistics I July 8, / 15

21 Distribution for Linear Combinations Liang Zhang (UofU) Applied Statistics I July 8, / 15

22 Distribution for Linear Combinations Proposition (Continued) Let X 1, X 2,..., X n have mean values µ 1, µ 2,..., µ n, respectively, and variances σ 2 1, σ2 2,..., σ2 n, respectively. 3. More generally, for any X 1, X 2,..., X n V (a 1 X 1 + a 2 X a n X n ) = n n a i a j Cov(X i, X j ) i=1 j=1 Liang Zhang (UofU) Applied Statistics I July 8, / 15

23 Distribution for Linear Combinations Proposition (Continued) Let X 1, X 2,..., X n have mean values µ 1, µ 2,..., µ n, respectively, and variances σ 2 1, σ2 2,..., σ2 n, respectively. 3. More generally, for any X 1, X 2,..., X n V (a 1 X 1 + a 2 X a n X n ) = n n a i a j Cov(X i, X j ) i=1 j=1 We call a 1 X 1 + a 2 X a n X n a linear combination of the X i s. Liang Zhang (UofU) Applied Statistics I July 8, / 15

24 Distribution for Linear Combinations Liang Zhang (UofU) Applied Statistics I July 8, / 15

25 Distribution for Linear Combinations Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. Liang Zhang (UofU) Applied Statistics I July 8, / 15

26 Distribution for Linear Combinations Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? Liang Zhang (UofU) Applied Statistics I July 8, / 15

27 Distribution for Linear Combinations Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? b. What is the variance of your total waiting time? Liang Zhang (UofU) Applied Statistics I July 8, / 15

28 Distribution for Linear Combinations Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? b. What is the variance of your total waiting time? c. What are the expected value and variance of the difference between morning and evening waiting times on a given day? Liang Zhang (UofU) Applied Statistics I July 8, / 15

29 Distribution for Linear Combinations Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? b. What is the variance of your total waiting time? c. What are the expected value and variance of the difference between morning and evening waiting times on a given day? d. What are the expected value and variance of the difference between total morning waiting time and total evening waiting time on a particular week? Liang Zhang (UofU) Applied Statistics I July 8, / 15

30 Distribution for Linear Combinations Liang Zhang (UofU) Applied Statistics I July 8, / 15

31 Distribution for Linear Combinations Corollary E(X 1 X 2 ) = E(X 1 ) E(X 2 ) and, if X 1 and X 2 are independent, V (X 1 X 2 ) = V (X 1 ) + V (X 2 ). Liang Zhang (UofU) Applied Statistics I July 8, / 15

32 Distribution for Linear Combinations Corollary E(X 1 X 2 ) = E(X 1 ) E(X 2 ) and, if X 1 and X 2 are independent, V (X 1 X 2 ) = V (X 1 ) + V (X 2 ). Proposition If X 1, X 2,..., X n are independent, normally distributed rv s (with possibly different means and/or variances), then any linear combination of the X i s also has a normal distribution. In particular, the difference X 1 X 2 between two independent, normally distributed variables is itself normally distributed. Liang Zhang (UofU) Applied Statistics I July 8, / 15

33 Distribution for Linear Combinations Liang Zhang (UofU) Applied Statistics I July 8, / 15

34 Distribution for Linear Combinations Example (Problem 62) Manufacture of a certain component requires three different maching operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are 15, 30, and 20min, respectively, and the standard deviations are 1, 2, and 1.5min, respectively. Liang Zhang (UofU) Applied Statistics I July 8, / 15

35 Distribution for Linear Combinations Example (Problem 62) Manufacture of a certain component requires three different maching operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are 15, 30, and 20min, respectively, and the standard deviations are 1, 2, and 1.5min, respectively. What is the probability that it takes at most 1 hour of machining time to produce a randomly selected component? Liang Zhang (UofU) Applied Statistics I July 8, / 15

36 Point Estimation Liang Zhang (UofU) Applied Statistics I July 8, / 15

37 Point Estimation Example (a variant of Problem 62, Ch5) Manufacture of a certain component requires three different maching operations. The total time for manufacturing one such component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. If we did an experiment in which we manufactured 10 components and record the operation time, and the sample time is given as following: time time What can we say about the population mean µ and population variance σ 2? Liang Zhang (UofU) Applied Statistics I July 8, / 15

38 Point Estimation Liang Zhang (UofU) Applied Statistics I July 8, / 15

39 Point Estimation Example (a variant of Problem 64, Ch5) Suppose the waiting time for a certain bus in the morning is uniformly distributed on [0, θ], where θ is unknown. If we record 10 waiting times as follwos: time time What can we say about the parameter θ? Liang Zhang (UofU) Applied Statistics I July 8, / 15

40 Point Estimation Liang Zhang (UofU) Applied Statistics I July 8, / 15

41 Point Estimation Definition A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of θ. Liang Zhang (UofU) Applied Statistics I July 8, / 15

42 Point Estimation Definition A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of θ. e.g. X = 10 i=1 X i/10 is a point estimator for µ for the normal distribution example. Liang Zhang (UofU) Applied Statistics I July 8, / 15

43 Point Estimation Definition A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of θ. e.g. X = 10 i=1 X i/10 is a point estimator for µ for the normal distribution example. The largest sample data X 10,10 is a point estimator for θ for the uniform distribution example. Liang Zhang (UofU) Applied Statistics I July 8, / 15

44 Methods of Point Estimation Liang Zhang (UofU) Applied Statistics I July 8, / 15

45 Methods of Point Estimation Definition Let X 1, X 2,..., X n be a random sample from a distribution with pmf or pdf f (x). For k = 1, 2, 3,..., the kth population moment, or kth moment of the distribution f (x), is E(X k ). The kth sample moment is 1 n n i=1 X k i. Liang Zhang (UofU) Applied Statistics I July 8, / 15

46 Methods of Point Estimation Definition Let X 1, X 2,..., X n be a random sample from a distribution with pmf or pdf f (x). For k = 1, 2, 3,..., the kth population moment, or kth moment of the distribution f (x), is E(X k ). The kth sample moment is 1 n n i=1 X k i. Definition Let X 1, X 2,..., X n be a random sample from a distribution with pmf or pdf f (x; θ 1,..., θ m ), where θ 1,..., θ m are parameters whose values are unknown. Then the moment estimators ˆθ 1,..., ˆθ m are obtained by equating the first m sample moments to the corresponding first m population moments and solving for θ 1,..., θ m. Liang Zhang (UofU) Applied Statistics I July 8, / 15