1. How different is the t distribution from the normal?

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Statistics Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. t-distributions. Comparison of two means: pooling of estimates of variances, or paired observations. In Lecture 6, when discussing comparison of two Binomial proportions, I was content to estimate unknown variances when calculating statistics that were to be treated as approximately normally distributed. You might have worried about the effect of variability of the estimate. W. S. Gosset ( Student ) considered a similar problem in a very famous 1908 paper, where the role of Student s t-distribution was first recognized. Gosset discovered that the effect of estimated variances could be described exactly in a simplified problem where n independent observations X 1,...,X n are taken from a normal distribution, N(µ, σ ). The sample mean, X = (X X n )/n has a N(µ, σ/ n) distribution. The random variable Z = X µ σ/ n has a standard normal distribution. If we estimate σ 2 by the sample variance, s 2 = i (X i X) 2 /(n 1), then the resulting statistic, T = X µ s/ n no longer has a normal distribution. It has a t-distribution on n 1 degrees of freedom. Remark. I have written T, instead of the t used by M&M page 505. I find it causes confusion that t refers to both the name of the statistic and the name of its distribution. As you will soon see, the estimation of the variance has the effect of spreading out the distribution a little beyond what it would be if σ were used. The effect is more pronounced at small sample sizes. With larger n, the difference between the normal and the t-distribution is really not very important. It is much more important that you understand the logic behind the interpretation of Z than the mostly small modifications needed for the t-distribution. 1. How different is the t distribution from the normal? In each of the following nine pictures I have plotted the quantiles of various t-distributions (with degrees of freedom as noted on the vertical axis) against the corresponding quantiles of the standard normal distribution along the horizontal axis. The plots are the population analogs of the normal (probability or quantile whichever name you prefer) plots from M&M Chapter 1. Indeed, you could think of these pictures as normal plots for immense sample sizes from the t-distributions. The upward curve at the right-hand end and the downward curve at the left-hand end tell us that the extreme quantiles for the t-distributions lie further from zero than the corresponding standard normal quantiles. The tails of the t-distributions decrease more slowly than the tails of the normal distribution (compare with M&M page 506). The t-distributions are more spread out than the normal. The spreading effect is huge for 1 degree of freedom, as shown by the first plot in the first row, but you should not be too alarmed. It is, after all, a rather dubious proposition that one can get a reasonable measure of spread from only two observations on a normal distribution.

2 Statistics Lecture 7 (20 October 98) c David Pollard Page 2 t on 1 df t on 2 df t on 3 df t on 8 df t on 9 df t on 10 df t on 20 df t on 30 df t on 50 df With 8 degrees of freedom (a sample size of 9) the extra spread is noticeable only for normal quantiles beyond 2. That is, the differences only start to stand out when we consider the extreme 2.5% of the tail probability in either direction. The plots in the last row are practically straight lines. You are pretty safe (unless you are interested in the extreme tails) in ignoring the difference between the standard normal and the t-distribution when the degrees of freedom are even moderately large. The small differences between normal and t-distributions are perhaps easier to see in the next figure % 95% 90% degrees of freedom Quantiles for the t-distribution, plotted against degrees of freedom. Corresponding standard normal values shown as tickmarks on the right margin.

3 Statistics Lecture 7 (20 October 98) c David Pollard Page 3 The top curve plots the quantile q n (.975), that is value for which P{T q n (.975)} =0.975 when T has a t-distibution with n degrees of freedom, against n. The middle curve shows the corresponding q n (0.95) values, and the bottom curve the q n (0.90) values. The three small marks at the right-hand edge of the picture give the corresponding values for the standard normal distribution. For example, the top mark sits at 1.96, because only 2.5% of the normal distribution lies further than 1.96 standard deviations out in the right tail. Notice how close the values for the t-distributions are to the values for the normal, with even only 4 or five degrees of freedom. Remark. You might wonder why the curves in the previous figure are shown as continuous lines, and not just as a series of points for degrees of freedom n = 1, 2, 3,... The answer is that the t-distribution can be defined for non-integer values of n. You should ignore this Remark unless you start pondering the formula displayed near the top of M&M page 549. The approximation requires fractional degrees of freedom. <1> Example. From a sample of n independent observations on the N(µ, σ ) distribution, with known σ, the interval X ± 1.96σ/ n has probability 0.95 of containing the unknown µ; it is a 95% confidence interval. If σ is unknown, it can estimated by the sample standard deviation s. The 95% confidence interval has almost the same form as before, X ±Cs/ n, where the unknown standard deviation σ is replaced by the estimated standard deviation, and the constant 1.96 is replaced by a larger constant C, which depends on the sample size. For example, from Minitab (or from Table D at the back of M&M), C = if n 1 = 1, or C = 2.23 where n 1 = 10. If n is large, C The values of C correspond to the top curve in the figure showing tail probabilities for the t-distribution. 2. How important is the t distribution? In their book Data Analysis and Regression, two highly eminent statisticians, Frederick Mosteller and John Tukey, pointed out that the value of Student s work lay not in a great numerical change but rather in recognition that one could, if appropriate assumptions held, make allowances for the uncertainties of small samples, not only in Student s original problem but in others as well; provision of numerical assessment of how small the necessary numerical adjustments of confidence points were in Student s problem and...how they depended on the extremeness of the probabilities involved; presentation of tables that could be used...to assess the uncertainty associated with even very small samples. They also noted some drawbacks, such as it made it too easy to neglect the proviso if appropriate assumptions held ; it overemphasized the exactness of Student s solution for his idealized problem; it helped to divert attention of theoretical statisticians to the development of exact ways of treating other problems. (Quotes taken from pages 5 and 6 of the book.) Mosteller and Tukey also noted one more specific disadvantage, as well as making a number of other more general observations about the true value of Student s work. In short, the use of t-distributions, in place of normal approximations, is not as crucial as you might be led to believe from a reading of some text books. The big differences between the exact t-distribution and the normal approximation arise only

4 Statistics Lecture 7 (20 October 98) c David Pollard Page 4 for quite small sample sizes. Moreover the derivation of the appealing exactness of the t-distribution holds only in the idealized setting of perfect normality. In situations where normality of the data is just an approximation, and where sample sizes are not extremely small, I feel it is unwise to fuss about the supposed precision of t-distributions. 3. Matched pairs In his famous paper, Gosset illustrated the use of his discovery by reference to a set of data on the number of hours of additional sleep gained by patients when treated with the dextro- and laevo- forms (optical isomers) of the drug hyoscyamine hydrobromide. patient dextro laevo difference For each patient, one would expect some dependence between the reponses to the two isomers. It would be unwise to treat the dextro- and laevo- values for a single patient as independent. However, randomization lets us act as if the differences in the effects correspond to 10 independent observations, X 1,...,X 10, on some N(µ, σ ) distribution. ( I have not checked the original source of the data, so I do not know whether the order in which the treatments was applied was randomized, as it should have been.) If there were no real difference between the effects of the two isomers we would have µ = 0. In that case, X would have a N(0,σ 2 /10) distribution, and T = X/ s 2 /10 would have a t-distribution on 9 degrees of freedom. The observed differences have sample mean equal to 1.58 and sample standard deviation equal to The observed value of T equals 1.58/(1.23/ 10) 4.06 Under the hypothesis that µ = 0, there is a probability of about of obtaining a value for T that is equal to 4.06 or larger. It is implausible that the differences in the effect are just random fluctuations. The laevo- isomer does seem to produce more sleep. Remark. One of the observed differences, 4.6, is much larger than the others. The value of 0 for the dextro- isomer with patient 9 also seems a little suspicious. Without knowing more about the data set, I would be wary about discarding the values for patient 9. I would point out, however, that without patient 9, the sample mean drops to 1.24 and the sample standard error drops to The single large value made the differences have a much larger spread. The value of T increases to 5.66, pushing the p-value (for a t-distribution on 8 degrees of freedom) down to Comparison of two sample means Suppose we have two independent sets of observations, X 1,...,X m from a N(µ X,σ X ), and Y 1,...,Y n from a N(µ Y,σ Y ). How should we test the hypothesis that µ X = µ Y?

5 Statistics Lecture 7 (20 October 98) c David Pollard Page 5 The two sample means are independent, with X distributed N(µ X,σ X / m) and Y distributed N(µ Y,σ Y / n). The difference X Y is distributed N(µ, τ), with µ = µ X µ Y and τ 2 = σ X 2 m + σ Y 2 n If σ X and σ y were known, we could compare (X Y )/τ with a standard normal, to test whether µ X = µ Y. If the variances were unknown, we could estimate τ 2 by τ 2 = s2 X m + s2 Y n. The statistic T = (X Y )/ τ would have approximately a standard normal distribution if µ X = µ Y. In general, it does not have exactly a t-distribution, even under the exact normality assumption. M&M pages describe various ways to refine the normal approximation things like using a t-distribution with a conservatively low choice of degrees of freedom, or using a t-distribution with estimated degrees of freedom. For reasonable sample sizes I tend to ignore such refinements. For example, I would have gone with the normal approximation in Example 7.16; I find the discussion of t-distributions in that example rather irrelevant (not to mention the unfortunate typo in the last line). Pooling If we know, or are prepared to assume, that σ X = σ Y = σ 2 then it is possible to recover an exact t-distribution for T under the null hypothesis µ X = µ Y, if we use a pooled estimator for the variance σ 2, and σ 2 pooled = (m 1)s2 X + (n 1)s2 Y m + n 2 ), instead of τ 2.Ifµ X =µ Y, then τ 2 pooled = σ 2 pooled ( 1 m + 1 n (X Y )/ τ pooled has a t-distribution on m + n 2 degrees of freedom. Read M&M for more about the risks of pooling. 5. Summary For testing hypotheses about means of normal distributions (or for setting confidence intervals), we have often to estimate unknown variances. With precisely normal data, the effect of estimation can be precisely determined in some situations. The consequence for those situations is that we have to adjust various constants, replacing values from the standard normal distribution by values from a t-distribution, with an appropriate number of degrees of freedom. When sample sizes are not very small, the adjustments tend to be small. You should use the procedures based on t-distributions, in programs like Minitab, when appropriate. But you should not get carried away with thought that you are carrying out exact inferences assumptions of normality are usually just convenient approximations.

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

2. Simple Linear Regression

Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

17. SIMPLE LINEAR REGRESSION II

17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

Need for Sampling. Very large populations Destructive testing Continuous production process

Chapter 4 Sampling and Estimation Need for Sampling Very large populations Destructive testing Continuous production process The objective of sampling is to draw a valid inference about a population. 4-

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

Chi Square Tests. Chapter 10. 10.1 Introduction

Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

Two-sample hypothesis testing, II 9.07 3/16/2004

Two-sample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For two-sample tests of the difference in mean, things get a little confusing, here,

An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

12: Analysis of Variance. Introduction

1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example 1. 2. We want to know if the mean battery lifespan exceeds the 300-minute

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

Two-sample inference: Continuous data

Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

Fixed-Effect Versus Random-Effects Models

CHAPTER 13 Fixed-Effect Versus Random-Effects Models Introduction Definition of a summary effect Estimating the summary effect Extreme effect size in a large study or a small study Confidence interval

AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

2 ESTIMATION. Objectives. 2.0 Introduction

2 ESTIMATION Chapter 2 Estimation Objectives After studying this chapter you should be able to calculate confidence intervals for the mean of a normal distribution with unknown variance; be able to calculate

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

Chapter 7 Section 1 Homework Set A

Chapter 7 Section 1 Homework Set A 7.15 Finding the critical value t *. What critical value t * from Table D (use software, go to the web and type t distribution applet) should be used to calculate the

Math 108 Exam 3 Solutions Spring 00

Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8

How To Run Statistical Tests in Excel

How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

Tests of Hypotheses Using Statistics

Tests of Hypotheses Using Statistics Adam Massey and Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract We present the various methods of hypothesis testing that one

Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

Confidence intervals

Confidence intervals Today, we re going to start talking about confidence intervals. We use confidence intervals as a tool in inferential statistics. What this means is that given some sample statistics,

Estimation and Confidence Intervals

Estimation and Confidence Intervals Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall Properties of Point Estimates 1 We have already encountered two point estimators: th e

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

Using Excel for inferential statistics

FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

9. Sampling Distributions

9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling

Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

The Standard Normal distribution

The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

Motor and Household Insurance: Pricing to Maximise Profit in a Competitive Market

Motor and Household Insurance: Pricing to Maximise Profit in a Competitive Market by Tom Wright, Partner, English Wright & Brockman 1. Introduction This paper describes one way in which statistical modelling

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

AP Statistics 2010 Scoring Guidelines

AP Statistics 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

August 2012 EXAMINATIONS Solution Part I

August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,

COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.

277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies

THE COST OF CAPITAL THE EFFECT OF CHANGES IN GEARING

December 2015 Examinations Chapter 19 Free lectures available for - click here THE COST OF CAPITAL THE EFFECT OF CHANGES IN GEARING 103 1 Introduction In this chapter we will look at the effect of gearing

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

Unit 26: Small Sample Inference for One Mean

Unit 26: Small Sample Inference for One Mean Prerequisites Students need the background on confidence intervals and significance tests covered in Units 24 and 25. Additional Topic Coverage Additional coverage

C. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.

Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample

AP Statistics 2002 Scoring Guidelines

AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought

Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

CHAPTER 13. Experimental Design and Analysis of Variance

CHAPTER 13 Experimental Design and Analysis of Variance CONTENTS STATISTICS IN PRACTICE: BURKE MARKETING SERVICES, INC. 13.1 AN INTRODUCTION TO EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE Data Collection

Least Squares Estimation

Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

Hypothesis Testing for Beginners

Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

STAT 350 Practice Final Exam Solution (Spring 2015)

PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

DATA INTERPRETATION AND STATISTICS

PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

Analysis of Variance ANOVA

Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.

Two Related Samples t Test

Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person

11. Analysis of Case-control Studies Logistic Regression

Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

CHI-SQUARE: TESTING FOR GOODNESS OF FIT

CHI-SQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity

An introduction to statistical data analysis (Summer 2014) Lecture notes. Taught by Shravan Vasishth [vasishth@uni-potsdam.de]

An introduction to statistical data analysis (Summer 2014) Lecture notes Taught by Shravan Vasishth [vasishth@uni-potsdam.de] Last edited: May 9, 2014 2 > sessioninfo() R version 3.0.2 (2013-09-25) Platform:

Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation

Understanding Confidence Intervals and Hypothesis Testing Using Excel Data Table Simulation Leslie Chandrakantha lchandra@jjay.cuny.edu Department of Mathematics & Computer Science John Jay College of

2013 MBA Jump Start Program. Statistics Module Part 3

2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just

Introduction to Analysis of Variance (ANOVA) Limitations of the t-test

Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

Objectives. 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) CI)

Objectives 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) Statistical confidence (CIS gives a good explanation of a 95% CI) Confidence intervals. Further reading http://onlinestatbook.com/2/estimation/confidence.html

Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

Unit 27: Comparing Two Means

Unit 27: Comparing Two Means Prerequisites Students should have experience with one-sample t-procedures before they begin this unit. That material is covered in Unit 26, Small Sample Inference for One

Principles of Hypothesis Testing for Public Health

Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions

Chapter 2 Probability Topics SPSS T tests

Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the One-Sample T test has been explained. In this handout, we also give the SPSS methods to perform

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

AP Statistics 2005 Scoring Guidelines

AP Statistics 2005 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the