, t, F distributions.

Slides:

Advertisements

Similar presentations

Introduction to the t Statistic

Advertisements

Chapter 23: Inferences About Means

Gossets student-t model What happens to quantitative samples when n is small?

Distributions of sampling statistics Chapter 6 Sample mean & sample variance.

The Simple Regression Model

Tests of Significance for Regression & Correlation b* will equal the population parameter of the slope rather thanbecause beta has another meaning with.

Chapter 27 Inferences for Regression This is just for one sample We want to talk about the relation between waist size and %body fat for the complete population.

One sample T Interval Example: speeding 90% confidence interval n=23 Check conditions Model: t n-1 Confidence interval: 31.0±1.52 = (29.48, 32.52) STAT.

Confidence Interval and Hypothesis Testing for:

10-1 Introduction 10-2 Inference for a Difference in Means of Two Normal Distributions, Variances Known Figure 10-1 Two independent populations.

Final Jeopardy $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 LosingConfidenceLosingConfidenceTesting.

Tuesday, October 22 Interval estimation. Independent samples t-test for the difference between two means. Matched samples t-test.

PROBABILITY AND SAMPLES: THE DISTRIBUTION OF SAMPLE MEANS.

Lecture 7 1 Statistics Statistics: 1. Model 2. Estimation 3. Hypothesis test.

Copyright © 2010 Pearson Education, Inc. Chapter 24 Comparing Means.

Some Continuous Probability Distributions Asmaa Yaseen.

Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal

Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.

X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ μ.

Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution.

Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 24 Comparing Means.

Measures of Variability Objective: Students should know what a variance and standard deviation are and for what type of data they typically used.

+ Chapter 12: Inference for Regression Inference for Linear Regression.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 9 Inferences Based on Two Samples.

Use of moment generating functions 1.Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …).

Experimental Design and Analysis Instructor: Mark Hancock February 8, 2008Slides by Mark Hancock.

Determination of Sample Size: A Review of Statistical Theory

Chapter 7 Sampling and Sampling Distributions ©. Simple Random Sample simple random sample Suppose that we want to select a sample of n objects from a.

CHAPTER SEVEN ESTIMATION. 7.1 A Point Estimate: A point estimate of some population parameter is a single value of a statistic (parameter space). For.

7 sum of RVs. 7-1: variance of Z Find the variance of Z = X+Y by using Var(X), Var(Y), and Cov(X,Y)

Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.

Ch8.2 Ch8.2 Population Mean Test Case I: A Normal Population With Known Null hypothesis: Test statistic value: Alternative Hypothesis Rejection Region.

Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

Statistics for Business and Economics 8 th Edition Chapter 7 Estimation: Single Population Copyright © 2013 Pearson Education, Inc. Publishing as Prentice.

_ z = X -  XX - Wow! We can use the z-distribution to test a hypothesis.

Statistics: Unlocking the Power of Data Lock 5 Section 6.4 Distribution of a Sample Mean.

Comparing Means Chapter 24. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-1 Business Statistics, 3e by Ken Black Chapter.

Comparing Two Means Ch. 13. Two-Sample t Interval for a Difference Between Two Means.

Chi Square Test for Goodness of Fit Determining if our sample fits the way it should be.

Clase 3. Parameters and statistics Population: hypothetical set of N “source” observations (N very large) Sample: a set of n observations from the source.

Introduction to the t statistic. Steps to calculate the denominator for the t-test 1. Calculate variance or SD s 2 = SS/n-1 2. Calculate the standard.

GOSSET, William Sealy How shall I deal with these small batches of brew?

DATA ANALYSIS AND MODEL BUILDING LECTURE 4 Prof. Roland Craigwell Department of Economics University of the West Indies Cave Hill Campus and Rebecca Gookool.

T-TEST. Outline  Introduction  T Distribution  Example cases  Test of Means-Single population  Test of difference of Means-Independent Samples 

AP Statistics Chapter 24 Comparing Means. Objectives: Two-sample t methods Two-Sample t Interval for the Difference Between Means Two-Sample t Test for.

Chapter 14 Single-Population Estimation. Population Statistics Population Statistics:  , usually unknown Using Sample Statistics to estimate population.

Sampling and Sampling Distributions

Theme 8. Major probability distributions

Chapter 23 Comparing Means.

Chapter 4. Inference about Process Quality

Math 4030 – 10b Inferences Concerning Variances: Hypothesis Testing

Sec. 7-5: Central Limit Theorem

Maximum likelihood estimation

Psychology 202a Advanced Psychological Statistics

Maximum Likelihood Estimation

Sample Mean Distributions

Inference on Mean, Var Unknown

t distribution Suppose Z ~ N(0,1) independent of X ~ χ2(n). Then,

Random Sampling Population Random sample: Statistics Point estimate

Problem: If I have a group of 100 applicants for a college summer program whose mean SAT-Verbal is 525, is this group of applicants “above national average”?

Chapter 23 Comparing Means.

Sampling Distribution

Sampling Distribution

The estimate of the proportion (“p-hat”) based on the sample can be a variety of values, and we don’t expect to get the same value every time, but the.

Chapter 24 Comparing Means Copyright © 2009 Pearson Education, Inc.

ECE 5345 Sums of RV’s and Long-Term Averages Confidence Intervals

#8 Some Standard Distributions

5: Introduction to estimation

1/2555 สมศักดิ์ ศิวดำรงพงศ์

Presentation transcript:

, t, F distributions

distribution t distribution F distribution 2

You don’t need to care about complicated math formula. Just see, and remember only what I marked with .

iid = independent and identically distributed ★ Random sample (r.s.) : A sample which are collected by independent sampling procedure from an identically distributed population. iid = independent and identically distributed

X\Y 1 2 T 1/16 1/8 1/4 1/2 1.0 X\Y 1 2 T 1/12 1/4 1/6 1/2 1/3 1.0 X\Y 1 2 3 T 1/16 1/8 1/4 1/2 1.0 X\Y 1 2 T 1/4 1/2 1.0

are (mutually) independent (a) are iid are (mutually) independent (a) (b) 6

7

8

9

: degree of freedom ( a parameter, related to number of var’s) 10

11

13

★ 14

★ 15

Â 2 ® ; n Â 2 ® ; n 1 3 5 8 15 24 ∞ 0.975 0.001 0.216 0.831 2.180 6.262 12.401 0.95 0.004 0.352 1.145 2.733 7.261 13.848 0.05 3.841 7.815 11.071 15.507 24.996 36.415 0.025 5.024 9.348 12.833 17.535 27.488 39.364 ∞ ∞ ∞

For large 17

Lorentzian Laplacian Normal Cauchy Double Expon. Gaussian

-4 -2 2 4 0.0 0.1 0.2 0.3 0.4 0.5

Louis F. Cauchy (1760–1848) Hendrik A. Lorentz (1853–1928)

21

: degree of freedom ( a parameter, related to number of var’s) 22

★ 23

★ Cauchy N(0,1) N(0,1) t(30) t(1) t(1), t(2), t(4), t(30) t(1) t(1) t(n t(n ) ) t( t( ∞ ∞ ) ) Cauchy N(0,1) 0.4 N(0,1) t(30) 0.3 0.2 t(1) 0.1 0.0 -4 -2 2 4 t(1), t(2), t(4), t(30)

0.3 1.0 0.8 0.2 0.6 0.4 0.1 0.2 0.0 0.0 -15 -10 -5 5 10 15 -15 -10 -5 5 10 15 1 3 8 15 21 23 ∞ 0.10 3.078 1.638 1.397 1.341 1.323 1.319 1.282 0.05 6.314 2.353 1.860 1.753 1.721 1.714 1.645 0.025 12.706 3.182 2.306 2.131 2.080 2.069 1.96

Student (1908), The probable error of a mean. Biometrika, 6-1, 1-25. William Sealy Gosset (1876-1937) Student (1908), The probable error of a mean. Biometrika, 6-1, 1-25.

: degree of freedom 28

Variance of F distribution is complicated. for large Variance of F distribution is complicated. You don’t need to care about these. 29

Most of cases the center of dist’n is near 1. 1 2 3 4 5 6 0.0 0.5 1.0 1.5 Most of cases the center of dist’n is near 1.

George W. Snedecor (1882 -1974) R. A. Fisher (1890 –1962)

32

1 2 3 4 5 6 7 0.1 0.3 0.5 0.7

1 2 6 8 9 15 161.45 18.513 5.987 5.318 5.117 4.543 199.50 19.000 5.143 4.459 4.256 3.682 233.99 19.330 4.284 3.581 3.374 2.790 238.88 19.371 4.147 3.438 3.230 2.641 240.54 19.385 4.099 3.388 3.179 2.588 245.95 19.429 3.938 3.218 3.006 2.403 1 2 6 8 9 15 647.79 38.506 8.813 7.571 7.209 6.200 799.50 39.000 7.260 6.059 5.715 4.765 937.11 39.331 5.820 4.652 4.320 3.415 956.66 39.373 5.600 4.433 4.102 3.199 963.29 39.387 5.523 4.357 4.026 3.123 984.87 39.431 5.269 4.101 3.769 2.862

★ 36

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0.358 3.94 0.254 2.79

38

-4 -2 2 4 2 4 6 8 -2.447 2.447

Thank you !!