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School
York University
Department
Course
Professor
Ying Kong
Semester
Fall

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CH10 Statistical inference -acquire information about populations from samples. -Estimation X  *Point Estimator-A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point. a) (single value –P(x)=0) virtually certain that the estimate will be wrong. b) often need to know how close the estimator is to the parameter. c) (sample n , Not reflect to Parameter value) intuitively reasonable to expect that a large sample will produce more accurate results, but point estimators don’t Type I Error  Reject H0(null hypothesis) when H0is true. P(Type I error) have the capacity to reflect the effects of larger sample sizes. = α (significance level)- when an innocent person is wrongly convicted *Interval estimator- An interval estimator draws inferences about a population by Type II Error  Do not reject H when H is false. P(Type II error) = 0 0 estimating the value of an unknown parameter using an interval.( range of values) β(Greek letter beta)- when a guilty defendant is acquitted Chapter 11- Hypotheses testing  and  Inversely related- any attempt to reduce one will increase the other. If the confidence is 95% then the significance is 5% (alpha).  Increasing significance level,     |     Four commonly used confidence levels  Increasing sample size n, n    The power of a test is defined as 1–   P(reject null hypothesis when it’s false)  Probability of Type II Error: E.g.  = 5%, n = 100,  = 10, H :  = 200, H :  # 200, given  = 203 0 1 Estimating the Population Mean   R.R.: x   x  200 x 198.04 X  Z  Z / 2  1.96 The width of the confidence interval is affected by  / 2  / n 10 / 100 • the population standard deviation σ) –SD, confidence interval n • the confidence level (1-α) –confidence level, confidence interval x   x  200 x  201.96  Z /2 1.96 • the sample size (n). –sample size decreases, the confidence interval.  / n 10 / 100  Unbiased Estimator-a population parameter is an estimator whose expected   P(198 .04  x  201 .96, given   203) value is equal to that parameter.X)  Consistency- if the difference between the estimator and the parameter grows   P(4.96  Z  1.04)  .5 .3508  .1492 smaller as the sample size grows larger. ( /n , n, variance) -4.96 -1.04 198 .04  203 201 .96  203  Relative Efficiency-If two unbiased estimators of a parameter, the one whose  P (  Z  ) variance is smaller is said to be relatively more efficient. (Smaller variance is better)10 / 100 10 / 100 Selecting the Sample size 2 Chapter 12: Inference about a Population when  is unknown  use t-test -estimate the mean to within B units X B  Z  /2  n     T distribution (σ is unknown, we use its point estimator s)  mound-shaped, Chapter 11 Hypothesis Test: to determine whether there is enough statistical symmetrical around zero. The “degrees of freedom” v/ d.f. =n-1, (a function of the samples)ze evidence in favor of a certain belief about a parameter determine how spread the distribution is (compared to the normal distribution) Hypothesis -Assume null H 0s true, find evident to infer al1. H  T-test only valid when the histogram is NOT extremely nonnormal. – Null Hypothesis (always “=“) – Alternative or research Hypothesis (Can be “”, or “=“) Test Statistic for μ When σ Is Unknown x   d.f.=n1 • Test Statistic z  x t  / n s / n Confidence interval estimating μ when σ is unknown • Rejection Region -a range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothX  t S d.f.=n1 tail RR Z -RR t -RR X -RR  / 2 n 2 2 2 > 1(+) right Z> Z  t >t ,n 1 X > X  ,n 1  Inference about a Population Variance σ < 1(-) left Z< t < 2 2 The sample variance s is an unbiased, consistent and efficient point estimator  Z   t ,n 1 X < X 1 ,n 1 for σ2 = 2( ) both Z> t > 2 2 Test Statistic for σ -Chisquared,  Z  / 2 t / 2 ,n 1 X > X  / 2 ,n 1 2 Z<  Z t < X < X 2 2 (n  1)s d.f.=n1  / 2  t  / 2 ,n 1 1 ( / 2 ),n 1   2 • P-Value (only work in Z distribution)-the probability of observing a test statistic  at least as extreme as the one computed given that the null hypothesis is true. P ( X 21 ( / 2 ) 2  X 2  / 2  1   Eg: The probability of observing a sample mean at least as large as 178 when =170. (n  1)s 2 2 (n  1)s 2 LCL     UCL    / 2 ,n 1  1 / 2 ,n 1 Inference about a Population Proportion (nomin
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