Chapter 4 STATISTICALINFERENCE ESTIMATION AND HYPOTHESES TESTING Statistical inference draws conclusions about a population /i.e, probability density function(PDF/from a random sample that has supposedly been drawn from that population ●●●●●●
Chapter 4 STATISTICAL INFERENCE: ESTIMATION AND HYPOTHESES TESTING Statistical inference draws conclusions about a population [i.e., probability density function (PDF) ] from a random sample that has supposedly been drawn from that population
4.1 THE MEANING OF STATISTICAL INFERENCE Statistical inference: the study of the relationship between a population and a sample drawn for that population The process of generalizing from the sample value ( to the population value E(X)is the essence of statistical inference
4.1 THE MEANING OF STATISTICAL INFERENCE Statistical inference: the study of the relationship between a population and a sample drawn for that population. The process of generalizing from the sample value ( ) to the population value E(X) is the essence of statistical inference. X
4.2 ESTIMATION AND HYPOTHESIS TESTING TWIN BRANCHES OF STATISTICAL INFERENCE 1. Estimation Estimation: the first step in statistical inference X: an estimator/statistic of the population parameter E(X), estimate: the particularnumerical value of the estimator sampling variation /sampling error: the variation in estimation from sample to sample. 2. Hypothesis testing In hypothesis testing we may have a prior judgment or expectation about what value a particularparameter may assume
4.2 ESTIMATION AND HYPOTHESIS TESTING: TWIN BRANCHES OF STATISTICAL INFERENCE 1. Estimation Estimation: the first step in statistical inference. : an estimator/statistic of the population parameter E(X), estimate: the particular numerical value of the estimator sampling variation /sampling error: the variation in estimation from sample to sample. 2.Hypothesis testing In hypothesis testing we may have a prior judgment or expectation about what value a particular parameter may assume. X
4.3 ESTIMATION OF PARAMETERS The usual procedure of estimation to assume that we have a random sample of size n from the known probability distribution and use the sample to estimate the unknown parameters, that is, use the sample mean as an estimate of the population mean (or expected value) and the sample variance as an estimate of the population variance 1. Point estimate A point estimator, or a statistic, is an r.V., its value will vary from sample to sample How can we rely on just one estimate X of the true population mean
4.3 ESTIMATION OF PARAMETERS The usual procedure of estimation: —— to assume that we have a random sample of size n from the known probability distribution and use the sample to estimate the unknown parameters, that is, use the sample mean as an estimate of the population mean (or expected value) and the sample variance as an estimate of the population variance. • 1. Point estimate A point estimator, or a statistic, is an r.v., its value will vary from sample to sample. How can we rely on just one estimate of the true population mean. X
2. Interval estimate Although x is the single"best guess of the true population mean, the interval, say, from 8 to 14, most likely includes the true u? This is interval estimation Sampling or probability distribution:X N(L, (X- N(21) P(tn1≤t≤tn1)=1-a S/vn P(X ≤Hx≤x+ critical t values:±t n-1 confidence interval X X (lower limit-upper limit confidence coefficient: 1-a level of significance/the prob of committing type I error: a
2. Interval estimate Although is the single “best” guess of the true population mean, the interval, say, from 8 to 14, most likely includes the true μχ ? This is interval estimation. Sampling or probability distribution: • • P(-t n-1 ≤t≤t n-1 )=1－α critical t values:±t n-1 confidence interval: (lower limit-upper limit) confidence coefficient: 1－α level of significance/the prob. of committing type I error: α X ~ ( , ) 2 n X N x ~ (0,1) / ( ) N n X Z X − = ~ ( 1) / − − = n X t S n X t 1 α n t S μ X n t S P(X n 1 X n 1 − + = − − − n t S X n t S X n X n−1 −1 − +