Dongfeng Li. Autumn 2010


 Garey Flynn
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1 Autumn 2010
2 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis testing, basic analysis of variance.
3 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis testing, basic analysis of variance.
4 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis testing, basic analysis of variance.
5 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis testing, basic analysis of variance.
6 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis testing, basic analysis of variance.
7 Section Contents Review statistics concepts: Distribution, discrete distribution, continuouse distribution, PDF, CDF, quantile. Normal distribution., median, variance, standard deviation, interquantile range, skewness, kurtosis. Population, parameter, sample, statistics, estimates, sampling distribution. MLE, standard error. Hypothesis tests, two types of errors, pvalue.
8 Section Contents Review statistics concepts: Distribution, discrete distribution, continuouse distribution, PDF, CDF, quantile. Normal distribution., median, variance, standard deviation, interquantile range, skewness, kurtosis. Population, parameter, sample, statistics, estimates, sampling distribution. MLE, standard error. Hypothesis tests, two types of errors, pvalue.
9 Section Contents Review statistics concepts: Distribution, discrete distribution, continuouse distribution, PDF, CDF, quantile. Normal distribution., median, variance, standard deviation, interquantile range, skewness, kurtosis. Population, parameter, sample, statistics, estimates, sampling distribution. MLE, standard error. Hypothesis tests, two types of errors, pvalue.
10 Section Contents Review statistics concepts: Distribution, discrete distribution, continuouse distribution, PDF, CDF, quantile. Normal distribution., median, variance, standard deviation, interquantile range, skewness, kurtosis. Population, parameter, sample, statistics, estimates, sampling distribution. MLE, standard error. Hypothesis tests, two types of errors, pvalue.
11 Section Contents Review statistics concepts: Distribution, discrete distribution, continuouse distribution, PDF, CDF, quantile. Normal distribution., median, variance, standard deviation, interquantile range, skewness, kurtosis. Population, parameter, sample, statistics, estimates, sampling distribution. MLE, standard error. Hypothesis tests, two types of errors, pvalue.
12 Section Contents Review statistics concepts: Distribution, discrete distribution, continuouse distribution, PDF, CDF, quantile. Normal distribution., median, variance, standard deviation, interquantile range, skewness, kurtosis. Population, parameter, sample, statistics, estimates, sampling distribution. MLE, standard error. Hypothesis tests, two types of errors, pvalue.
13 Distribution Random Variable: Discrete, such as sex, patient/control, age group. Continuous, such as weight, blood pressure. Distribution: used to describe the relative chance of taking some value. For discrete variable X, use P(X = xi ), where {x i } are the value set of X. Called probability mass function(pmf). For continuous variable X, use the probability density function(pdf) f (x), where P(X (x ɛ, x + ɛ) f (x)(2ɛ).
14 Distribution Random Variable: Discrete, such as sex, patient/control, age group. Continuous, such as weight, blood pressure. Distribution: used to describe the relative chance of taking some value. For discrete variable X, use P(X = xi ), where {x i } are the value set of X. Called probability mass function(pmf). For continuous variable X, use the probability density function(pdf) f (x), where P(X (x ɛ, x + ɛ) f (x)(2ɛ).
15 Distribution Random Variable: Discrete, such as sex, patient/control, age group. Continuous, such as weight, blood pressure. Distribution: used to describe the relative chance of taking some value. For discrete variable X, use P(X = xi ), where {x i } are the value set of X. Called probability mass function(pmf). For continuous variable X, use the probability density function(pdf) f (x), where P(X (x ɛ, x + ɛ) f (x)(2ɛ).
16 Distribution Random Variable: Discrete, such as sex, patient/control, age group. Continuous, such as weight, blood pressure. Distribution: used to describe the relative chance of taking some value. For discrete variable X, use P(X = xi ), where {x i } are the value set of X. Called probability mass function(pmf). For continuous variable X, use the probability density function(pdf) f (x), where P(X (x ɛ, x + ɛ) f (x)(2ɛ).
17 Distribution Random Variable: Discrete, such as sex, patient/control, age group. Continuous, such as weight, blood pressure. Distribution: used to describe the relative chance of taking some value. For discrete variable X, use P(X = xi ), where {x i } are the value set of X. Called probability mass function(pmf). For continuous variable X, use the probability density function(pdf) f (x), where P(X (x ɛ, x + ɛ) f (x)(2ɛ).
18 Distribution Random Variable: Discrete, such as sex, patient/control, age group. Continuous, such as weight, blood pressure. Distribution: used to describe the relative chance of taking some value. For discrete variable X, use P(X = xi ), where {x i } are the value set of X. Called probability mass function(pmf). For continuous variable X, use the probability density function(pdf) f (x), where P(X (x ɛ, x + ɛ) f (x)(2ɛ).
19 CDF Cumulative distribution function(cdf) F(x): F(x) = P(X x) P(x (a, b]) = F(b) F(a) For discrete distribution, F(x) = P(X = x i ) x i x For continuous distribution, F(x) = x f (t)dt
20 CDF Cumulative distribution function(cdf) F(x): F(x) = P(X x) P(x (a, b]) = F(b) F(a) For discrete distribution, F(x) = P(X = x i ) x i x For continuous distribution, F(x) = x f (t)dt
21 CDF Cumulative distribution function(cdf) F(x): F(x) = P(X x) P(x (a, b]) = F(b) F(a) For discrete distribution, F(x) = P(X = x i ) x i x For continuous distribution, F(x) = x f (t)dt
22 Quantile function Quantile function: the inverse of CDF q(p) = F 1 (p), p (0, 1) if F (x) is 11 mapping. Generally, q(p) = x p where P(X x p ) p, P(X x p ) 1 p (x p can be nonunique.)
23 Quantile function Quantile function: the inverse of CDF q(p) = F 1 (p), p (0, 1) if F (x) is 11 mapping. Generally, q(p) = x p where P(X x p ) p, P(X x p ) 1 p (x p can be nonunique.)
24 The normal distribution Standard normal distribution(n(0,1)), PDF ϕ(x) = 1 2π e x2 2 Normal distribution N(µ, σ 2 ), PDF f (x) = ϕ( x µ σ ) = 1 2πσ exp{ (x µ)2 2σ 2 }
25 The normal distribution Standard normal distribution(n(0,1)), PDF ϕ(x) = 1 2π e x2 2 Normal distribution N(µ, σ 2 ), PDF f (x) = ϕ( x µ σ ) = 1 2πσ exp{ (x µ)2 2σ 2 }
26 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
27 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
28 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
29 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
30 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
31 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
32 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
33 Numerical charasteristics PDF is a curve with infinite number of points. We can use some numbers to describe the key part of a distribution. Firstly, the location measurement. The mean EX. The meadian, x0.5 = q(0.5) where (can be nonunique). Variability measurement. P(x x 0.5 ) 0.5, P(x x 0.5 ) 0.5 The standard deviation σ X = E(X EX) 2. Interquantile range q(0.75) q(0.25).
34 Shape characteristics. Skewness ( X EX E σ X ) 3 Symmetric, leftskewed, rightskewed density. Shape characteristics. Kurtosis ( ) X EX 4 E 3 σ X Heavy tail problem. Multimodel distribution. E.g., the weight of 10 year old and 20 year old mixed together.
35 Shape characteristics. Skewness ( X EX E σ X ) 3 Symmetric, leftskewed, rightskewed density. Shape characteristics. Kurtosis ( ) X EX 4 E 3 σ X Heavy tail problem. Multimodel distribution. E.g., the weight of 10 year old and 20 year old mixed together.
36 Shape characteristics. Skewness ( X EX E σ X ) 3 Symmetric, leftskewed, rightskewed density. Shape characteristics. Kurtosis ( ) X EX 4 E 3 σ X Heavy tail problem. Multimodel distribution. E.g., the weight of 10 year old and 20 year old mixed together.
37 Shape characteristics. Skewness ( X EX E σ X ) 3 Symmetric, leftskewed, rightskewed density. Shape characteristics. Kurtosis ( ) X EX 4 E 3 σ X Heavy tail problem. Multimodel distribution. E.g., the weight of 10 year old and 20 year old mixed together.
38 Shape characteristics. Skewness ( X EX E σ X ) 3 Symmetric, leftskewed, rightskewed density. Shape characteristics. Kurtosis ( ) X EX 4 E 3 σ X Heavy tail problem. Multimodel distribution. E.g., the weight of 10 year old and 20 year old mixed together.
39 Population A population is modeled by a random variable or a distribution. Population parameters: unknown numbers which could decide the distribution. E.g., for N(µ, σ 2 ) population, (µ, σ 2 ) are the two unknown population parameters.
40 Population A population is modeled by a random variable or a distribution. Population parameters: unknown numbers which could decide the distribution. E.g., for N(µ, σ 2 ) population, (µ, σ 2 ) are the two unknown population parameters.
41 Sample A sample(typically) is n observations draw independently from the population, X 1, X 2,..., X n. A sample can be regarded as n numbers, or n iid random variables. Statistics: calculate some values to estimate the distribution of the population. Can be regarded as a random variable. Such as sample mean, sample standard deviation. Can be used to estimate unknown population parameters, called estimates. Sampling distribution: The distribution of an estimator or other statistic. E.g., if X 1,..., X n is a sample from N(µ, σ 2 ), then X N(µ, σ 2 /n).
42 Sample A sample(typically) is n observations draw independently from the population, X 1, X 2,..., X n. A sample can be regarded as n numbers, or n iid random variables. Statistics: calculate some values to estimate the distribution of the population. Can be regarded as a random variable. Such as sample mean, sample standard deviation. Can be used to estimate unknown population parameters, called estimates. Sampling distribution: The distribution of an estimator or other statistic. E.g., if X 1,..., X n is a sample from N(µ, σ 2 ), then X N(µ, σ 2 /n).
43 Sample A sample(typically) is n observations draw independently from the population, X 1, X 2,..., X n. A sample can be regarded as n numbers, or n iid random variables. Statistics: calculate some values to estimate the distribution of the population. Can be regarded as a random variable. Such as sample mean, sample standard deviation. Can be used to estimate unknown population parameters, called estimates. Sampling distribution: The distribution of an estimator or other statistic. E.g., if X 1,..., X n is a sample from N(µ, σ 2 ), then X N(µ, σ 2 /n).
44 Sample A sample(typically) is n observations draw independently from the population, X 1, X 2,..., X n. A sample can be regarded as n numbers, or n iid random variables. Statistics: calculate some values to estimate the distribution of the population. Can be regarded as a random variable. Such as sample mean, sample standard deviation. Can be used to estimate unknown population parameters, called estimates. Sampling distribution: The distribution of an estimator or other statistic. E.g., if X 1,..., X n is a sample from N(µ, σ 2 ), then X N(µ, σ 2 /n).
45 MLE(Maximimum Likelihood Estimate) MLE is a commonly used parameter estimation method. For random sample X 1, X 2,..., X n from population X, if PDF of PMF of X is f (x; β), then ˆβ = arg max β L(β) = arg max β n f (X i ; β) i=1 is called the MLE of the unknow parameter β. Under some conditions, when the sample size n, ˆβ has a limiting(approximate) distribution N(β, σ 2 β ).
46 MLE(Maximimum Likelihood Estimate) MLE is a commonly used parameter estimation method. For random sample X 1, X 2,..., X n from population X, if PDF of PMF of X is f (x; β), then ˆβ = arg max β L(β) = arg max β n f (X i ; β) i=1 is called the MLE of the unknow parameter β. Under some conditions, when the sample size n, ˆβ has a limiting(approximate) distribution N(β, σ 2 β ).
47 Standard Error If ˆβ = ψ(x 1,..., X n ) is a estimate of an population parameter β, it has a sampling distribution F ˆβ(x). The standard deviation of the sampling distribution is σ ˆβ. An estimate of σ ˆβ is called the standard error(se) of ˆβ. If ˆβ has a limiting normal distribution, then ˆβ is approximately distributed N(β, SE( ˆβ)). SE can measure the precision of estimation. SE can be used to construct (approximate) confidence intervals.
48 Standard Error If ˆβ = ψ(x 1,..., X n ) is a estimate of an population parameter β, it has a sampling distribution F ˆβ(x). The standard deviation of the sampling distribution is σ ˆβ. An estimate of σ ˆβ is called the standard error(se) of ˆβ. If ˆβ has a limiting normal distribution, then ˆβ is approximately distributed N(β, SE( ˆβ)). SE can measure the precision of estimation. SE can be used to construct (approximate) confidence intervals.
49 Standard Error If ˆβ = ψ(x 1,..., X n ) is a estimate of an population parameter β, it has a sampling distribution F ˆβ(x). The standard deviation of the sampling distribution is σ ˆβ. An estimate of σ ˆβ is called the standard error(se) of ˆβ. If ˆβ has a limiting normal distribution, then ˆβ is approximately distributed N(β, SE( ˆβ)). SE can measure the precision of estimation. SE can be used to construct (approximate) confidence intervals.
50 Standard Error If ˆβ = ψ(x 1,..., X n ) is a estimate of an population parameter β, it has a sampling distribution F ˆβ(x). The standard deviation of the sampling distribution is σ ˆβ. An estimate of σ ˆβ is called the standard error(se) of ˆβ. If ˆβ has a limiting normal distribution, then ˆβ is approximately distributed N(β, SE( ˆβ)). SE can measure the precision of estimation. SE can be used to construct (approximate) confidence intervals.
51 Standard Error If ˆβ = ψ(x 1,..., X n ) is a estimate of an population parameter β, it has a sampling distribution F ˆβ(x). The standard deviation of the sampling distribution is σ ˆβ. An estimate of σ ˆβ is called the standard error(se) of ˆβ. If ˆβ has a limiting normal distribution, then ˆβ is approximately distributed N(β, SE( ˆβ)). SE can measure the precision of estimation. SE can be used to construct (approximate) confidence intervals.
52 Standard Error If ˆβ = ψ(x 1,..., X n ) is a estimate of an population parameter β, it has a sampling distribution F ˆβ(x). The standard deviation of the sampling distribution is σ ˆβ. An estimate of σ ˆβ is called the standard error(se) of ˆβ. If ˆβ has a limiting normal distribution, then ˆβ is approximately distributed N(β, SE( ˆβ)). SE can measure the precision of estimation. SE can be used to construct (approximate) confidence intervals.
53 Hypothesis Tests To test the null hypothesis H 0 agaist the alternative hypothesis H a on the population; Given sample X 1, X 2,..., X n from population F(x; θ), construct some statistic ξ, whose distribution does not depend on θ, but its value can indicate the possible choice of H 0 or H a. Traditionally, we first choose an significance level α, then we find a rejection area W, where sup H0 P(ξ W ) α. We reject H 0 when ξ W.
54 Hypothesis Tests To test the null hypothesis H 0 agaist the alternative hypothesis H a on the population; Given sample X 1, X 2,..., X n from population F(x; θ), construct some statistic ξ, whose distribution does not depend on θ, but its value can indicate the possible choice of H 0 or H a. Traditionally, we first choose an significance level α, then we find a rejection area W, where sup H0 P(ξ W ) α. We reject H 0 when ξ W.
55 Hypothesis Tests To test the null hypothesis H 0 agaist the alternative hypothesis H a on the population; Given sample X 1, X 2,..., X n from population F(x; θ), construct some statistic ξ, whose distribution does not depend on θ, but its value can indicate the possible choice of H 0 or H a. Traditionally, we first choose an significance level α, then we find a rejection area W, where sup H0 P(ξ W ) α. We reject H 0 when ξ W.
56 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
57 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
58 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
59 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
60 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
61 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
62 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
63 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
64 Errors and PValues Two types of possible errors in a hypothesis test: Type I error, H 0 true but rejected. Error rate is at most the significance level α. Type II error, H0 false but accepted. Error rate can be as large as 1 α. To reduce type II error: Construct theoretically good tests. Don t choose α too small. Choose a big enough sample size n. pvalue: the minimum α we can use if we want to reject H 0, after the test statistic value is known. The smaller the pvalue is, the more confidence we have when we reject H 0. Reject H 0 if and only if the pvalue is less than α.
65 Example hypothesis test X N(µ, σ 2 ). Sample is X 1,..., X n. H 0 : µ µ 0 H a : µ > µ 0. Test statistic T = X µ 0 SE( X) where SE( X) = S/ n, S is the sample standard deviation. Rejection area: W = {ξ > λ}, λ = F 1 (1 α, n 1), F 1 (p, n) is the quantile function of the t(n 1) distribution. Pvalue is 1 F(T ; n 1), where F(x; n) is the CDF of the t(n) distribution. The larger T is, the smaller the pvalue is, the more confidence we have when we reject H 0.
66 Example hypothesis test X N(µ, σ 2 ). Sample is X 1,..., X n. H 0 : µ µ 0 H a : µ > µ 0. Test statistic T = X µ 0 SE( X) where SE( X) = S/ n, S is the sample standard deviation. Rejection area: W = {ξ > λ}, λ = F 1 (1 α, n 1), F 1 (p, n) is the quantile function of the t(n 1) distribution. Pvalue is 1 F(T ; n 1), where F(x; n) is the CDF of the t(n) distribution. The larger T is, the smaller the pvalue is, the more confidence we have when we reject H 0.
67 Example hypothesis test X N(µ, σ 2 ). Sample is X 1,..., X n. H 0 : µ µ 0 H a : µ > µ 0. Test statistic T = X µ 0 SE( X) where SE( X) = S/ n, S is the sample standard deviation. Rejection area: W = {ξ > λ}, λ = F 1 (1 α, n 1), F 1 (p, n) is the quantile function of the t(n 1) distribution. Pvalue is 1 F(T ; n 1), where F(x; n) is the CDF of the t(n) distribution. The larger T is, the smaller the pvalue is, the more confidence we have when we reject H 0.
68 Example hypothesis test X N(µ, σ 2 ). Sample is X 1,..., X n. H 0 : µ µ 0 H a : µ > µ 0. Test statistic T = X µ 0 SE( X) where SE( X) = S/ n, S is the sample standard deviation. Rejection area: W = {ξ > λ}, λ = F 1 (1 α, n 1), F 1 (p, n) is the quantile function of the t(n 1) distribution. Pvalue is 1 F(T ; n 1), where F(x; n) is the CDF of the t(n) distribution. The larger T is, the smaller the pvalue is, the more confidence we have when we reject H 0.
69 Example hypothesis test X N(µ, σ 2 ). Sample is X 1,..., X n. H 0 : µ µ 0 H a : µ > µ 0. Test statistic T = X µ 0 SE( X) where SE( X) = S/ n, S is the sample standard deviation. Rejection area: W = {ξ > λ}, λ = F 1 (1 α, n 1), F 1 (p, n) is the quantile function of the t(n 1) distribution. Pvalue is 1 F(T ; n 1), where F(x; n) is the CDF of the t(n) distribution. The larger T is, the smaller the pvalue is, the more confidence we have when we reject H 0.
70 Example hypothesis test X N(µ, σ 2 ). Sample is X 1,..., X n. H 0 : µ µ 0 H a : µ > µ 0. Test statistic T = X µ 0 SE( X) where SE( X) = S/ n, S is the sample standard deviation. Rejection area: W = {ξ > λ}, λ = F 1 (1 α, n 1), F 1 (p, n) is the quantile function of the t(n 1) distribution. Pvalue is 1 F(T ; n 1), where F(x; n) is the CDF of the t(n) distribution. The larger T is, the smaller the pvalue is, the more confidence we have when we reject H 0.
71 Section Contents Measurment level of variables. statistics for nominal variables. statistics for interval variables. Histograms, boxplots, QQ plots, probability plots, stemleaf plots. Using PROC FREQ, PROC MEANS, PROC UNIVARIATE. statistics Summary statistics
72 Section Contents Measurment level of variables. statistics for nominal variables. statistics for interval variables. Histograms, boxplots, QQ plots, probability plots, stemleaf plots. Using PROC FREQ, PROC MEANS, PROC UNIVARIATE. statistics Summary statistics
73 Section Contents Measurment level of variables. statistics for nominal variables. statistics for interval variables. Histograms, boxplots, QQ plots, probability plots, stemleaf plots. Using PROC FREQ, PROC MEANS, PROC UNIVARIATE. statistics Summary statistics
74 Section Contents Measurment level of variables. statistics for nominal variables. statistics for interval variables. Histograms, boxplots, QQ plots, probability plots, stemleaf plots. Using PROC FREQ, PROC MEANS, PROC UNIVARIATE. statistics Summary statistics
75 Section Contents Measurment level of variables. statistics for nominal variables. statistics for interval variables. Histograms, boxplots, QQ plots, probability plots, stemleaf plots. Using PROC FREQ, PROC MEANS, PROC UNIVARIATE. statistics Summary statistics
76 Measurement levels Nominal, such as sex, job type, race. Ordinal, such as age group(child, youth, medium, old), dose level(none, low, high). Interval, such as weight, blood pressure, heart rate. statistics Summary statistics
77 Measurement levels Nominal, such as sex, job type, race. Ordinal, such as age group(child, youth, medium, old), dose level(none, low, high). Interval, such as weight, blood pressure, heart rate. statistics Summary statistics
78 Measurement levels Nominal, such as sex, job type, race. Ordinal, such as age group(child, youth, medium, old), dose level(none, low, high). Interval, such as weight, blood pressure, heart rate. statistics Summary statistics
79 Statistics for nominal variables The value set. The count of each value(called frequency), and the percentage. Use a bar chart to show the distribution. statistics Summary statistics
80 Statistics for nominal variables The value set. The count of each value(called frequency), and the percentage. Use a bar chart to show the distribution. statistics Summary statistics
81 Statistics for nominal variables The value set. The count of each value(called frequency), and the percentage. Use a bar chart to show the distribution. statistics Summary statistics
82 Example: frequency table Use PROC FREQ to list the value set and the frequency counts, percents. proc freq data=sasuser.class; tables sex; run; Gender sex Frequency Percent Cumulative Cumulative Frequency Percent F M statistics Summary statistics
83 Example: bar chart Use PROC GCHART to make a bar chart. hbar could be replaced with vbar, hbar3d, vbar3d. proc gchart data=sasuser.class; hbar sex; run; statistics Summary statistics
84 statistics for interval variables Location statistics: sample mean, sample median. Variability statistics: sample standard deviation, sample interquantile range, range, coefficient of variation. Sample skewness Sample kurtosis Moments, quantiles. n (n 1)(n 2) n(n+1) (n 1)(n 2)(n 3) ( ) y i y 3. s (yi y) 4 3(n 1)2 s 4 (n 2)(n 3) statistics Summary statistics
85 statistics for interval variables Location statistics: sample mean, sample median. Variability statistics: sample standard deviation, sample interquantile range, range, coefficient of variation. Sample skewness Sample kurtosis Moments, quantiles. n (n 1)(n 2) n(n+1) (n 1)(n 2)(n 3) ( ) y i y 3. s (yi y) 4 3(n 1)2 s 4 (n 2)(n 3) statistics Summary statistics
86 statistics for interval variables Location statistics: sample mean, sample median. Variability statistics: sample standard deviation, sample interquantile range, range, coefficient of variation. Sample skewness Sample kurtosis Moments, quantiles. n (n 1)(n 2) n(n+1) (n 1)(n 2)(n 3) ( ) y i y 3. s (yi y) 4 3(n 1)2 s 4 (n 2)(n 3) statistics Summary statistics
87 statistics for interval variables Location statistics: sample mean, sample median. Variability statistics: sample standard deviation, sample interquantile range, range, coefficient of variation. Sample skewness Sample kurtosis Moments, quantiles. n (n 1)(n 2) n(n+1) (n 1)(n 2)(n 3) ( ) y i y 3. s (yi y) 4 3(n 1)2 s 4 (n 2)(n 3) statistics Summary statistics
88 statistics for interval variables Location statistics: sample mean, sample median. Variability statistics: sample standard deviation, sample interquantile range, range, coefficient of variation. Sample skewness Sample kurtosis Moments, quantiles. n (n 1)(n 2) n(n+1) (n 1)(n 2)(n 3) ( ) y i y 3. s (yi y) 4 3(n 1)2 s 4 (n 2)(n 3) statistics Summary statistics
89 PROC UNIVARIATE Use PROC UNIVARIATE to compute sample statistics for an interval variable. Example: proc univariate data=sasuser.class; var height; run; statistics Summary statistics
90 PROC UNIVARIATE Use PROC UNIVARIATE to compute sample statistics for an interval variable. Example: proc univariate data=sasuser.class; var height; run; statistics Summary statistics
91 Output Moments N 19 Sum Weights Sum Observations Std Deviation Skewness Kurtosis Uncorrected SS Corrected SS Coeff Variation Std Error Basic Statistical Measures Location Variability Std Deviation Median Mode Range Interquartile Range statistics Summary statistics Note The mode displayed is the smallest of 2 modes with a count of 2.
92 Tests for Location: Mu0=0 Test Statistic p Value Student s t t Pr > t <.0001 Sign M 9.5 Pr >= M <.0001 Signed Rank S 95 Pr >= S <.0001 statistics Summary statistics
93 Quantiles (Definition 5) Quantile Estimate 100% Max % % % % Q % Median % Q % % % % Min 51.3 statistics Summary statistics
94 Extreme Observations Lowest Highest Value Obs Value Obs statistics Summary statistics
95 Histograms The histogram is a nonparametric estimate of the population density function. It divide the value set into intervals, and count the percent of observations in each interval. Example: proc univariate data=sasuser.class noprint; var height; histogram; run; statistics Summary statistics
96 Histograms The histogram is a nonparametric estimate of the population density function. It divide the value set into intervals, and count the percent of observations in each interval. Example: proc univariate data=sasuser.class noprint; var height; histogram; run; statistics Summary statistics
97 Histograms The histogram is a nonparametric estimate of the population density function. It divide the value set into intervals, and count the percent of observations in each interval. Example: proc univariate data=sasuser.class noprint; var height; histogram; run; statistics Summary statistics
98 Examples of histograms Skewness shown in histograms. statistics Summary statistics
99 Box plot Quantiles could be used to describe key distribution charasteristics. The box plot shows the median, 1/4 and 3/4 quantile, and the minimum and maximum of a variable. The box holds the middle 50% range. Length is the IQR(inter quantile range). The wiskers extend to the minimum and the maximum; but the length of the wisker is not allowed to exceed 1.5 times the IQR. Points beyond wiskers are plotted as individual points. They are candidate for outliers. statistics Summary statistics
100 Box plot Quantiles could be used to describe key distribution charasteristics. The box plot shows the median, 1/4 and 3/4 quantile, and the minimum and maximum of a variable. The box holds the middle 50% range. Length is the IQR(inter quantile range). The wiskers extend to the minimum and the maximum; but the length of the wisker is not allowed to exceed 1.5 times the IQR. Points beyond wiskers are plotted as individual points. They are candidate for outliers. statistics Summary statistics
101 Box plot Quantiles could be used to describe key distribution charasteristics. The box plot shows the median, 1/4 and 3/4 quantile, and the minimum and maximum of a variable. The box holds the middle 50% range. Length is the IQR(inter quantile range). The wiskers extend to the minimum and the maximum; but the length of the wisker is not allowed to exceed 1.5 times the IQR. Points beyond wiskers are plotted as individual points. They are candidate for outliers. statistics Summary statistics
102 Box plot Quantiles could be used to describe key distribution charasteristics. The box plot shows the median, 1/4 and 3/4 quantile, and the minimum and maximum of a variable. The box holds the middle 50% range. Length is the IQR(inter quantile range). The wiskers extend to the minimum and the maximum; but the length of the wisker is not allowed to exceed 1.5 times the IQR. Points beyond wiskers are plotted as individual points. They are candidate for outliers. statistics Summary statistics
103 Box plot Quantiles could be used to describe key distribution charasteristics. The box plot shows the median, 1/4 and 3/4 quantile, and the minimum and maximum of a variable. The box holds the middle 50% range. Length is the IQR(inter quantile range). The wiskers extend to the minimum and the maximum; but the length of the wisker is not allowed to exceed 1.5 times the IQR. Points beyond wiskers are plotted as individual points. They are candidate for outliers. statistics Summary statistics
104 Example: boxplot of height statistics Summary statistics
105 proc sql; create view work._tmp_0 as select height, 1 as _dummy_ from sasuser.class; title ; axis1 major=none value=none label=none; proc boxplot data=_tmp_0; plot height*_dummy_ / boxstyle=skematic cboxfill=blue haxis=axis1; run; statistics Summary statistics
106 Example: boxplot of GPA and CO Skewness and possible outliers shown in boxplot. statistics Summary statistics
107 Grouped boxplot statistics Summary statistics
108 proc sort data=sasuser.gpa out=_tmp_1; by sex; proc boxplot data=_tmp_1; plot gpa*sex / boxstyle=skematic cboxfill=blue; run; statistics Summary statistics
109 QQ Plot The normal distribution is the most commonly used distribution. Graphs are designed to show discrepency from the normal distribution. QQ plot is one of them. Suppose Y 1, Y 2,..., Y n is from N(µ,σ 2 ) and sorted ascendingly. CDF F(x) = Φ( x µ σ ), F(Y i) i n, Φ( Y i µ σ ) i n, Y i µ + σφ 1 ( i n ). Let x i = Φ 1 ( i n ), plot (x i, Y i ),i = 1,..., n. The points should lie around the line y = µ + σx. Continuity adjustment: x i = Φ 1 ( i n+0.25 ). statistics Summary statistics
110 QQ Plot The normal distribution is the most commonly used distribution. Graphs are designed to show discrepency from the normal distribution. QQ plot is one of them. Suppose Y 1, Y 2,..., Y n is from N(µ,σ 2 ) and sorted ascendingly. CDF F(x) = Φ( x µ σ ), F(Y i) i n, Φ( Y i µ σ ) i n, Y i µ + σφ 1 ( i n ). Let x i = Φ 1 ( i n ), plot (x i, Y i ),i = 1,..., n. The points should lie around the line y = µ + σx. Continuity adjustment: x i = Φ 1 ( i n+0.25 ). statistics Summary statistics
111 QQ Plot The normal distribution is the most commonly used distribution. Graphs are designed to show discrepency from the normal distribution. QQ plot is one of them. Suppose Y 1, Y 2,..., Y n is from N(µ,σ 2 ) and sorted ascendingly. CDF F(x) = Φ( x µ σ ), F(Y i) i n, Φ( Y i µ σ ) i n, Y i µ + σφ 1 ( i n ). Let x i = Φ 1 ( i n ), plot (x i, Y i ),i = 1,..., n. The points should lie around the line y = µ + σx. Continuity adjustment: x i = Φ 1 ( i n+0.25 ). statistics Summary statistics
112 QQ Plot The normal distribution is the most commonly used distribution. Graphs are designed to show discrepency from the normal distribution. QQ plot is one of them. Suppose Y 1, Y 2,..., Y n is from N(µ,σ 2 ) and sorted ascendingly. CDF F(x) = Φ( x µ σ ), F(Y i) i n, Φ( Y i µ σ ) i n, Y i µ + σφ 1 ( i n ). Let x i = Φ 1 ( i n ), plot (x i, Y i ),i = 1,..., n. The points should lie around the line y = µ + σx. Continuity adjustment: x i = Φ 1 ( i n+0.25 ). statistics Summary statistics
113 QQ Plot The normal distribution is the most commonly used distribution. Graphs are designed to show discrepency from the normal distribution. QQ plot is one of them. Suppose Y 1, Y 2,..., Y n is from N(µ,σ 2 ) and sorted ascendingly. CDF F(x) = Φ( x µ σ ), F(Y i) i n, Φ( Y i µ σ ) i n, Y i µ + σφ 1 ( i n ). Let x i = Φ 1 ( i n ), plot (x i, Y i ),i = 1,..., n. The points should lie around the line y = µ + σx. Continuity adjustment: x i = Φ 1 ( i n+0.25 ). statistics Summary statistics
114 QQ Plot The normal distribution is the most commonly used distribution. Graphs are designed to show discrepency from the normal distribution. QQ plot is one of them. Suppose Y 1, Y 2,..., Y n is from N(µ,σ 2 ) and sorted ascendingly. CDF F(x) = Φ( x µ σ ), F(Y i) i n, Φ( Y i µ σ ) i n, Y i µ + σφ 1 ( i n ). Let x i = Φ 1 ( i n ), plot (x i, Y i ),i = 1,..., n. The points should lie around the line y = µ + σx. Continuity adjustment: x i = Φ 1 ( i n+0.25 ). statistics Summary statistics
115 Example: QQ Plot of HEIGHT proc univariate data=sasuser.class noprint; qqplot height / normal(mu=est sigma=est); run; statistics Summary statistics
116 Example: Skewness in QQ Plot statistics Summary statistics
117 Probability Plot Probability plot is the same plot as a QQ plot, exept that the label of the x axis Φ(x i ) instead of x i. Probability plots are preferable for graphical estimation of percentiles, whereas QQ plots are preferable for graphical estimation of distribution parameters. statistics Summary statistics
118 Probability Plot Probability plot is the same plot as a QQ plot, exept that the label of the x axis Φ(x i ) instead of x i. Probability plots are preferable for graphical estimation of percentiles, whereas QQ plots are preferable for graphical estimation of distribution parameters. statistics Summary statistics
119 Example: Probability Plot of HEIGHT proc univariate data=sasuser.class noprint; probplot height / normal(mu=est sigma=est); run; statistics Summary statistics
120 The Stemleaf Plot The stemleaf plot is a textbased plot, which display information like the histogram, but with detail on each data value. Each leaf corresponds to one data value. PROC UNIVARIATE has an option PLOT, which generates textbased stemleaf plot, boxplot and QQ plot. statistics Summary statistics
121 The Stemleaf Plot The stemleaf plot is a textbased plot, which display information like the histogram, but with detail on each data value. Each leaf corresponds to one data value. PROC UNIVARIATE has an option PLOT, which generates textbased stemleaf plot, boxplot and QQ plot. statistics Summary statistics
122 proc univariate data=sasuser.class plot; var height; run; Stem Leaf # Multiply Stem.Leaf by 10**+1 statistics Summary statistics
123 PROC MEANS and PROC SUMMARY PROC MEANS and PROC SUMMARY are used to produce summary statistics. They can display overall summary statistics and classified summary statistics. PROC MEANS displays result by default; PROC SUMMARY need the PRINT option to display result. They can output data sets with the summary statistics. PROC SUMMARY is designed to do this, although PROC MEANS can do the same. statistics Summary statistics
124 PROC MEANS and PROC SUMMARY PROC MEANS and PROC SUMMARY are used to produce summary statistics. They can display overall summary statistics and classified summary statistics. PROC MEANS displays result by default; PROC SUMMARY need the PRINT option to display result. They can output data sets with the summary statistics. PROC SUMMARY is designed to do this, although PROC MEANS can do the same. statistics Summary statistics
125 PROC MEANS and PROC SUMMARY PROC MEANS and PROC SUMMARY are used to produce summary statistics. They can display overall summary statistics and classified summary statistics. PROC MEANS displays result by default; PROC SUMMARY need the PRINT option to display result. They can output data sets with the summary statistics. PROC SUMMARY is designed to do this, although PROC MEANS can do the same. statistics Summary statistics
126 Example of overall statistics Use the VAR statement to specify which varibles to summarize. Example of overall statistics: proc means data=sasuser.class; var height weight; run; proc summary data=sasuser.class print; var height weight; run; statistics Summary statistics
127 Example of overall statistics Use the VAR statement to specify which varibles to summarize. Example of overall statistics: proc means data=sasuser.class; var height weight; run; proc summary data=sasuser.class print; var height weight; run; statistics Summary statistics
128 Classified summary Use the CLASS statement to specify one or more class variables. Example: proc means data=sasuser.class ; var height weight; class sex; run; statistics Summary statistics
129 Classified summary Use the CLASS statement to specify one or more class variables. Example: proc means data=sasuser.class ; var height weight; class sex; run; statistics Summary statistics
130 Other statistical measures PROC MEANS calculate summaries N,, Standard Deviation, Minimum, Maximum by default. Use PROC MEANS options to specify other statistical measures. Example: proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; run; statistics Summary statistics
131 Other statistical measures PROC MEANS calculate summaries N,, Standard Deviation, Minimum, Maximum by default. Use PROC MEANS options to specify other statistical measures. Example: proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; run; statistics Summary statistics
132 Other statistical measures PROC MEANS calculate summaries N,, Standard Deviation, Minimum, Maximum by default. Use PROC MEANS options to specify other statistical measures. Example: proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; run; statistics Summary statistics
133 Producing output datasets Use the OUTPUT statement to save the summary statistics to an output dataset. Syntax: OUTPUT OUT=outputdataset keyword= keyword= / AUTONAME; Where keyword is a name of some statistical measure, like MEAN, STD, N, MIN, MAX, etc. Example: statistics Summary statistics proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; output out=res N= CV= / AUTONAME; run; proc print data=res;run;
134 Producing output datasets Use the OUTPUT statement to save the summary statistics to an output dataset. Syntax: OUTPUT OUT=outputdataset keyword= keyword= / AUTONAME; Where keyword is a name of some statistical measure, like MEAN, STD, N, MIN, MAX, etc. Example: statistics Summary statistics proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; output out=res N= CV= / AUTONAME; run; proc print data=res;run;
135 Producing output datasets Use the OUTPUT statement to save the summary statistics to an output dataset. Syntax: OUTPUT OUT=outputdataset keyword= keyword= / AUTONAME; Where keyword is a name of some statistical measure, like MEAN, STD, N, MIN, MAX, etc. Example: statistics Summary statistics proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; output out=res N= CV= / AUTONAME; run; proc print data=res;run;
136 Producing output datasets Use the OUTPUT statement to save the summary statistics to an output dataset. Syntax: OUTPUT OUT=outputdataset keyword= keyword= / AUTONAME; Where keyword is a name of some statistical measure, like MEAN, STD, N, MIN, MAX, etc. Example: statistics Summary statistics proc means data=sasuser.class MEAN VAR CV; var height weight; class sex; output out=res N= CV= / AUTONAME; run; proc print data=res;run;
137 One sample Z tests, t tests. Two sample t tests. The Wilcoxon rank sum test. Paired t tests. Comparing proportions: one sample and two sample. One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
138 One sample Z tests, t tests. Two sample t tests. The Wilcoxon rank sum test. Paired t tests. Comparing proportions: one sample and two sample. One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
139 One sample Z tests, t tests. Two sample t tests. The Wilcoxon rank sum test. Paired t tests. Comparing proportions: one sample and two sample. One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
140 One sample Z tests, t tests. Two sample t tests. The Wilcoxon rank sum test. Paired t tests. Comparing proportions: one sample and two sample. One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
141 One sample Z tests, t tests. Two sample t tests. The Wilcoxon rank sum test. Paired t tests. Comparing proportions: one sample and two sample. One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
142 One Sample Z Test Twosided X N(µ, σ 2 0 ), σ 0 known. H 0 : µ = µ 0 H a : µ µ 0 Z = X µ 0 σ 0 / H 0 n N(0, 1). W = { Z > λ}, λ = Φ 1 (1 α/2). pvalue=2(1 Φ( Z ). Wald test: Based on the central limit theorem, to test H 0 : θ = θ 0 H a : θ θ 0, use W = ˆθ θ 0 as the SE(ˆθ) test statistic, if W N(0, 1). One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
143 One Sample Z Test Twosided X N(µ, σ 2 0 ), σ 0 known. H 0 : µ = µ 0 H a : µ µ 0 Z = X µ 0 σ 0 / H 0 n N(0, 1). W = { Z > λ}, λ = Φ 1 (1 α/2). pvalue=2(1 Φ( Z ). Wald test: Based on the central limit theorem, to test H 0 : θ = θ 0 H a : θ θ 0, use W = ˆθ θ 0 as the SE(ˆθ) test statistic, if W N(0, 1). One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
144 One Sample Z Test Twosided X N(µ, σ 2 0 ), σ 0 known. H 0 : µ = µ 0 H a : µ µ 0 Z = X µ 0 σ 0 / H 0 n N(0, 1). W = { Z > λ}, λ = Φ 1 (1 α/2). pvalue=2(1 Φ( Z ). Wald test: Based on the central limit theorem, to test H 0 : θ = θ 0 H a : θ θ 0, use W = ˆθ θ 0 as the SE(ˆθ) test statistic, if W N(0, 1). One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
145 One Sample Z Test Twosided X N(µ, σ 2 0 ), σ 0 known. H 0 : µ = µ 0 H a : µ µ 0 Z = X µ 0 σ 0 / H 0 n N(0, 1). W = { Z > λ}, λ = Φ 1 (1 α/2). pvalue=2(1 Φ( Z ). Wald test: Based on the central limit theorem, to test H 0 : θ = θ 0 H a : θ θ 0, use W = ˆθ θ 0 as the SE(ˆθ) test statistic, if W N(0, 1). One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
146 One Sample Z Test Twosided X N(µ, σ 2 0 ), σ 0 known. H 0 : µ = µ 0 H a : µ µ 0 Z = X µ 0 σ 0 / H 0 n N(0, 1). W = { Z > λ}, λ = Φ 1 (1 α/2). pvalue=2(1 Φ( Z ). Wald test: Based on the central limit theorem, to test H 0 : θ = θ 0 H a : θ θ 0, use W = ˆθ θ 0 as the SE(ˆθ) test statistic, if W N(0, 1). One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
147 One Sample Z Test Twosided X N(µ, σ 2 0 ), σ 0 known. H 0 : µ = µ 0 H a : µ µ 0 Z = X µ 0 σ 0 / H 0 n N(0, 1). W = { Z > λ}, λ = Φ 1 (1 α/2). pvalue=2(1 Φ( Z ). Wald test: Based on the central limit theorem, to test H 0 : θ = θ 0 H a : θ θ 0, use W = ˆθ θ 0 as the SE(ˆθ) test statistic, if W N(0, 1). One Sample Test of the Comparing Two Groups Paired Comparison Comparing Proportions
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