# Sampling and Hypothesis Testing

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus statistic. size mean variance proportion Population: N µ σ 2 π Sample: n x s 2 p 1 Properties of estimators: sample mean x = 1 n To make inferences regarding the population mean, µ, we need to know something about the probability distribution of this sample statistic, x. n i=1 x i If the sampling procedure is unbiased, deviations of x from µ in the upward and downward directions should be equally likely; on average, they should cancel out. E( x) = µ = E(X) The sample mean is then an unbiased estimator of the population mean. The distribution of a sample statistic is known as a sampling distribution. Two of its characteristics are of particular interest, the mean or expected value and the variance or standard deviation. E( x): Thought experiment: Sample repeatedly from the given population, each time recording the sample mean, and take the average of those sample means. 2 3

2 Efficiency One estimator is more efficient than another if its values are more tightly clustered around its expected value. E.g. alternative estimators for the population mean: x versus the average of the largest and smallest values in the sample. The degree of dispersion of an estimator is generally measured by the standard deviation of its probability distribution (sampling distribution). This goes under the name standard error. Standard error of sample mean σ x = σ n The more widely dispersed are the population values around their mean (larger σ ), the greater the scope for sampling error (i.e. drawing by chance an unrepresentative sample whose mean differs substantially from µ). A larger sample size (greater n) narrows the dispersion of x. 4 5 Other statistics Population proportion, π. The corresponding sample statistic is the proportion of the sample having the characteristic in question, p. The sample proportion is an unbiased estimator of the population proportion E(p) = π Its standard error is given by σ p = π(1 π) n Population variance, σ 2. σ 2 = 1 N (x i µ) 2 N i=1 Estimator, sample variance: s 2 = 1 n (x i x) 2 n 1 i=1 6 7

3 Shape of sampling distributions Besides knowing expected value and standard error, we also need to know the shape of a sampling distribution in order to put it to use. Not all sampling distributions are Gaussian, e.g. sample variance as estimator of population variance. In this case the ratio (n 1)s 2 /σ 2 follows a skewed distribution known as χ 2, (chi-square ) with n 1 degrees of freedom. Sample mean: Central Limit Theorem implies a Gaussian distribution, for large enough samples. Reminder: Five dice 0.2 P( x) If the sample size is large the χ 2 distribution converges towards the normal x 8 9 Confidence intervals If we know the mean, standard error and shape of the distribution of a given sample statistic, we can then make definite probability statements about the statistic. Example: µ = 100 and σ = 12 for a certain population, and we draw a sample with n = 36 from that population. The standard error of x is σ / n = 12/6 = 2, and a sample size of 36 is large enough to justify the assumption of a Gaussian sampling distribution. We know that the range µ ± 2σ encloses the central 95 percent of a normal distribution, so we can state P(96 < x < 104).95 There s a 95 percent probability that the sample mean lies within 4 units (= 2 standard errors) of the population mean, 100. Population mean unknown If µ is unknown we can still say P(µ 4 < x < µ + 4).95 With probability.95 the sample mean will be drawn from within 4 units of the unknown population mean. We go ahead and draw the sample, and calculate a sample mean of (say) 97. If there s a probability of.95 that our x came from within 4 units of µ, we can turn that around: we re entitled to be 95 percent confident that µ lies between 93 and 101. We draw up a 95 percent confidence interval for the population mean as x ± 2σ x

4 Population variance unknown With σ unknown, we have to estimate the standard error of x. s x ˆσ x = s n We can now reformulate our 95 percent confidence interval for µ: x ± 2s x. Strictly speaking, the substitution of s for σ alters the shape of the sampling distribution. Instead of being Gaussian it now follows the t distribution, which looks very much like the Gaussian except that it s a bit fatter in the tails. The t distribution Unlike the Gaussian, the t distribution is not fully characterized by its mean and standard deviation: there is an additional factor, namely the degrees of freedom (df). For estimating a population mean the df term is the sample size minus 1. At low degrees of freedom the t distribution is noticeably more dispersed than the Gaussian, meaning that a 95 percent confidence interval would have to be wider (greater uncertainty). As the degrees of freedom increase, the t distribution converges towards the Gaussian. Values enclosing the central 95 percent of the distribution: Normal: µ ± 1.960σ t(30): µ ± 2.042σ Further examples The following information regarding the Gaussian distribution enables you to construct a 99 percent confidence interval. P(µ 2.58σ < x < µ σ ) 0.99 Thus the 99 percent interval is x ± 2.58σ x. If we want greater confidence that our interval straddles the unknown parameter value (99 percent versus 95 percent) then our interval must be wider (±2.58 standard errors versus ±2 standard errors). Estimating a proportion An opinion polling agency questions a sample of 1200 people to assess the degree of support for candidate X. Sample info: p = Our single best guess at the population proportion, π, is then 0.56, but we can quantify our uncertainty. The standard error of p is π(1 π)/n. The value of π is unknown but we can substitute p or, to be conservative, we can put π = 0.5 which maximizes the value of π(1 π). On the latter procedure, the estimated standard error is 0.25/1200 = The large sample justifies the Gaussian assumption for the sampling distribution; the 95 percent confidence interval is 0.56 ± = 0.56 ±

5 Generalizing the idea Let θ denote a generic parameter. 1. Find an estimator (preferably unbiased) for θ. 2. Generate ˆθ (point estimate). 3. Set confidence level, 1 α. 4. Form interval estimate (assuming symmetrical distribution): ˆθ ± maximum error for (1 α) confidence z-scores Suppose the sampling distribution of ˆθ is Gaussian. The following notation is useful: z = x µ σ The standard normal score or z-score expresses the value of a variable in terms of its distance from the mean, measured in standard deviations. Example: µ = 1000 and σ = 50. The value x = 850 has a z-score of 3.0: it lies 3 standard deviations below the mean. Where the distribution of ˆθ is Gaussian we can write the 1 α confidence interval for θ as Maximum error equals so many standard errors of such and such a size. The number of standard errors depends on the chosen confidence level (possibly also the degrees of freedom). The size of the standard error, σˆθ, depends on the nature of the parameter being estimated and the sample size. 16 ˆθ ± σˆθ z α/2 17 The logic of hypothesis testing 0.95 Analogy between the set-up of a hypothesis test and a court of law. Defendant on trial in the statistical court is the null hypothesis, some definite claim regarding a parameter of interest. z.975 = 1.96 z.025 = 1.96 This is about as far as we can go in general terms. The specific formula for σˆθ depends on the parameter. Just as the defendant is presumed innocent until proved guilty, the null hypothesis (H 0 ) is assumed true (at least for the sake of argument) until the evidence goes against it. H 0 is in fact: Decision: True False Reject Type I error Correct decision P = α Fail to reject Correct decision Type II error P = β 1 β is the power of a test; trade-off between α and β

6 Choosing the significance level How do we get to choose α (probability of Type I error)? The calculations that compose a hypothesis test are condensed in a key number, namely a conditional probability: the probability of observing the given sample data, on the assumption that the null hypothesis is true. Say we use a cutoff of.01: we ll reject the null hypothesis if the p-value for the test is.01. If the null hypothesis is in fact true, what is the probability of our rejecting it? It s the probability of getting a p-value less than or equal to.01, which is (by definition).01. In selecting our cutoff we selected α, the probability of Type I error. This is called the p-value. If it is small, we can place one of two interpretations on the situation: (a) The null hypothesis is true and the sample we drew is an improbable, unrepresentative one. (b) The null hypothesis is false. The smaller the p-value, the less comfortable we are with alternative (a). (Digression) To reach a conclusion we must specify the limit of our comfort zone, a p-value below which we ll reject H Example of hypothesis test A maker of RAM chips claims an average access time of 60 nanoseconds (ns) for the chips. Quality control has the job of checking that the production process is maintaining acceptable access speed: they test a sample of chips each day. Today s sample information is that with 100 chips tested, the mean access time is 63 ns with a standard deviation of 2 ns. Is this an acceptable result? Sould we go with the symmetrical hypotheses H 0 : µ = 60 versus H 1 : µ 60? Well, we don t mind if the chips are faster than advertised. So instead we adopt the asymmetrical hypotheses: The p-value is where n = 100 and s = 2. P( x 63 µ 60) If the null hypothesis is true, E( x) is no greater than 60. The estimated standard error of x is s/ n = 2/10 =.2. With n = 100 we can take the sampling distribution to be normal. With a Gaussian sampling distribution the test statistic is the z-score. z = x µ H = = 15 s x.2 H 0 : µ 60 versus H 1 : µ > 60 Let α =

7 Variations on the example Suppose the test were as described above, except that the sample was of size 10 instead of 100. Given the small sample and the fact that the population standard deviation, σ, is unknown, we could not justify the assumption of a Gaussian sampling distribution for x. Rather, we d have to use the t distribution with df = 9. The estimated standard error, s x = 2/ 10 = 0.632, and the test statistic is t(9) = x µ H = s x.632 = 4.74 In general the test statistic can be written as ˆθ θ H0 test = That is, sample statistic minus the value stated in the null hypothesis which by assumption equals E(ˆθ) divided by the (estimated) standard error of ˆθ. The distribution to which test must be referred, in order to obtain the p-value, depends on the situation. sˆθ The p-value for this statistic is a lot larger than for z = 15, but still much smaller than the chosen significance level of 5 percent, so we still reject the null hypothesis Another variation We chose an asymmetrical test setup above. What difference would it make if we went with the symmetrical version, H 0 : µ = 60. Two-tailed test. Both high and low values count against H 0. H 0 : µ = 60 versus H 1 : µ 60? α/2 α/2 We have to think: what sort of values of the test statistic should count against the null hypothesis? In the asymmetrical case only values of x greater than 60 counted against H 0. A sample mean of (say) 57 would be consistent with µ 60; it is not even prima facie evidence against the null. Therefore the critical region of the sampling distribution (the region containing values that would cause us to reject the null) lies strictly in the upper tail. But if the null hypothesis were µ = 60, then values of x both substantially below and substantially above 60 would count against it. The critical region would be divided into two portions, one in each tail of the sampling distribution. 26 H 0 : µ 60. One-tailed test. Only high values count against H α

8 Practical consequence We must double the p-value, before comparing it to α. The sample mean was 63, and the p-value was defined as the probability of drawing a sample like this or worse, from the standpoint of H 0. In the symmetrical case, like this or worse means with a sample mean this far away from the hypothesized population mean, or farther, in either direction. More on p-values Let E denote the sample evidence and H denote the null hypothesis that is on trial. The p-value can then be expressed as P(E H). This may seem awkward. Wouldn t it be better to calculate the conditional probability the other way round, P(H E)? Instead of working with the probability of obtaining a sample like the one we in fact obtained, assuming the null hypothesis to be true, why can t we think in terms of the probability that the null hypothesis is true, given the sample evidence? So the p-value is P( x 63 x 57), which is double the value we found previously Recall the multiplication rule for probabilities, which we wrote as P(A B) = P(A) P(B A) Swapping the positions of A and B we can equally well write P(B A) = P(B) P(A B) And taking these two equations together we can infer that P(A) P(B A) = P(B) P(A B) Substituting E (Evidence) and H (null Hypothesis) for A and B, we get P(H E) = P(H) P(E H) P(E) We know how to find the p-value, P(E H). To obtain the probability we re now canvassing as an alternative, P(H E), we have to supply in addition P(H) and P(E). P(H) is the marginal probability of the null hypothesis and P(E) is the marginal probability of the sample evidence. Where are these going to come from?? or P(B A) = P(B) P(A B) P(A) This is Bayes rule. It provides a means of converting from a conditional probability one way round to the inverse conditional probability

9 Confidence intervals and tests The symbol α is used for both the significance level of a hypothesis test (the probability of Type I error), and in denoting the confidence level (1 α) for interval estimation. There is an equivalence between a two-tailed hypothesis test at significance level α and an interval estimate using confidence level 1 α. Suppose µ is unknown and a sample of size 64 yields x = 50, s = 10. The 95 percent confidence interval for µ is then In a two-tailed test at the 5 percent significance level, we fail to reject H 0 if and only if x falls within the central 95 percent of the sampling distribution, according to H 0. But since 55 exceeds 50 by more than the maximum error, 2.45, we can see that, conversely, the central 95 percent of a sampling distribution centered on 55 will not include 50, so a finding of x = 50 must lead to rejection of the null. Significance level and confidence level are complementary. ( ) ± 1.96 = 50 ± 2.45 = to Suppose we want to test H 0 : µ = 55 using the 5 percent significance level. No additional calculation is needed. The value 55 lies outside of the 95 percent confidence interval, so we can conclude that H 0 is rejected Digression normal p-value normal pdf The further we are from the center of the sampling distribution, according to H 0, the smaller the p-value. Back to the main discussion. 34

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

### Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### 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

### Chapter 9, Part A Hypothesis Tests. Learning objectives

Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Chapter 16 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 16 Multiple Choice Questions (The answers are provided after the last question.) 1. Which of the following symbols represents a population parameter? a. SD b. σ c. r d. 0 2. If you drew all possible

### Null Hypothesis H 0. The null hypothesis (denoted by H 0

Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

### Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### 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,

### [Chapter 10. Hypothesis Testing]

[Chapter 10. Hypothesis Testing] 10.1 Introduction 10.2 Elements of a Statistical Test 10.3 Common Large-Sample Tests 10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests

### MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample

MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set 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

### 15.0 More Hypothesis Testing

15.0 More Hypothesis Testing 1 Answer Questions Type I and Type II Error Power Calculation Bayesian Hypothesis Testing 15.1 Type I and Type II Error In the philosophy of hypothesis testing, the null hypothesis

### Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities

Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and

### Hypothesis Testing or How to Decide to Decide Edpsy 580

Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide

### Introduction to Hypothesis Testing OPRE 6301

Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about

### 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

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Chapter 8 Introduction to Hypothesis Testing

Chapter 8 Student Lecture Notes 8-1 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate

### Chapter III. Testing Hypotheses

Chapter III Testing Hypotheses R (Introduction) A statistical hypothesis is an assumption about a population parameter This assumption may or may not be true The best way to determine whether a statistical

### The Basics of a Hypothesis Test

Overview The Basics of a Test Dr Tom Ilvento Department of Food and Resource Economics Alternative way to make inferences from a sample to the Population is via a Test A hypothesis test is based upon A

### Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

### 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

### Hypothesis testing. c 2014, Jeffrey S. Simonoff 1

Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there

### 9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### Confidence intervals, t tests, P values

Confidence intervals, t tests, P values Joe Felsenstein Department of Genome Sciences and Department of Biology Confidence intervals, t tests, P values p.1/31 Normality Everybody believes in the normal

### Lesson 9 Hypothesis Testing

Lesson 9 Hypothesis Testing Outline Logic for Hypothesis Testing Critical Value Alpha (α) -level.05 -level.01 One-Tail versus Two-Tail Tests -critical values for both alpha levels Logic for Hypothesis

### Hypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic.

Hypothesis Testing Hypothesis testing allows us to use a sample to decide between two statements made about a Population characteristic. Population Characteristics are things like The mean of a population

### 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

### Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

### 9.1 Basic Principles of Hypothesis Testing

9. Basic Principles of Hypothesis Testing Basic Idea Through an Example: On the very first day of class I gave the example of tossing a coin times, and what you might conclude about the fairness of the

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### 22. HYPOTHESIS TESTING

22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING

CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized

### Extending Hypothesis Testing. p-values & confidence intervals

Extending Hypothesis Testing p-values & confidence intervals So far: how to state a question in the form of two hypotheses (null and alternative), how to assess the data, how to answer the question by

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### 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

### Statistics 641 - EXAM II - 1999 through 2003

Statistics 641 - EXAM II - 1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the

### Hypothesis Testing - II

-3σ -2σ +σ +2σ +3σ Hypothesis Testing - II Lecture 9 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in Probability & Statistics in Engineering Today in P&S -3σ -2σ +σ +2σ +3σ Review: Hypothesis

### Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

### Hypothesis Testing. Concept of Hypothesis Testing

Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population

### Estimation; Sampling; The T distribution

Estimation; Sampling; The T distribution I. Estimation A. In most statistical studies, the population parameters are unknown and must be estimated. Therefore, developing methods for estimating as accurately

### 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

### Chapter 9: Hypothesis Testing GBS221, Class April 15, 2013 Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College

Chapter Objectives 1. Learn how to formulate and test hypotheses about a population mean and a population proportion. 2. Be able to use an Excel worksheet to conduct hypothesis tests about population means

### Multiple random variables

Multiple random variables Multiple random variables We essentially always consider multiple random variables at once. The key concepts: Joint, conditional and marginal distributions, and independence of

### Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212.

Variance of OLS Estimators and Hypothesis Testing Charlie Gibbons ARE 212 Spring 2011 Randomness in the model Considering the model what is random? Y = X β + ɛ, β is a parameter and not random, X may be

### 1 Hypothesis Testing. H 0 : population parameter = hypothesized value:

1 Hypothesis Testing In Statistics, a hypothesis proposes a model for the world. Then we look at the data. If the data are consistent with that model, we have no reason to disbelieve the hypothesis. Data

### Hypothesis Testing Summary

Hypothesis Testing Summary Hypothesis testing begins with the drawing of a sample and calculating its characteristics (aka, statistics ). A statistical test (a specific form of a hypothesis test) is an

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Measuring the Power of a Test

Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection

### HYPOTHESIS TESTING AND TYPE I AND TYPE II ERROR

HYPOTHESIS TESTING AND TYPE I AND TYPE II ERROR Hypothesis is a conjecture (an inferring) about one or more population parameters. Null Hypothesis (H 0 ) is a statement of no difference or no relationship

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### Sampling Distributions and the Central Limit Theorem

135 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 10 Sampling Distributions and the Central Limit Theorem In the previous chapter we explained

### THE LOGIC OF HYPOTHESIS TESTING. The general process of hypothesis testing remains constant from one situation to another.

THE LOGIC OF HYPOTHESIS TESTING Hypothesis testing is a statistical procedure that allows researchers to use sample to draw inferences about the population of interest. It is the most commonly used inferential

### Hypothesis Testing --- One Mean

Hypothesis Testing --- One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis

### Estimation of σ 2, the variance of ɛ

Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated

### Chapter 1 Hypothesis Testing

Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population,

### 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

### Basic concepts and introduction to statistical inference

Basic concepts and introduction to statistical inference Anna Helga Jonsdottir Gunnar Stefansson Sigrun Helga Lund University of Iceland (UI) Basic concepts 1 / 19 A review of concepts Basic concepts Confidence

### 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.

### Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### Water Quality Problem. Hypothesis Testing of Means. Water Quality Example. Water Quality Example. Water quality example. Water Quality Example

Water Quality Problem Hypothesis Testing of Means Dr. Tom Ilvento FREC 408 Suppose I am concerned about the quality of drinking water for people who use wells in a particular geographic area I will test

### 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

### Chapter 8. Professor Tim Busken. April 20, Chapter 8. Tim Busken. 8.2 Basics of. Hypothesis Testing. Works Cited

Chapter 8 Professor April 20, 2014 In Chapter 8, we continue our study of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample

### 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

### 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

### General Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test

Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics

### Hypothesis Testing. Chapter Introduction

Contents 9 Hypothesis Testing 553 9.1 Introduction............................ 553 9.2 Hypothesis Test for a Mean................... 557 9.2.1 Steps in Hypothesis Testing............... 557 9.2.2 Diagrammatic

### 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

### Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### STT 430/630/ES 760 Lecture Notes: Chapter 6: Hypothesis Testing 1. February 23, 2009 Chapter 6: Introduction to Hypothesis Testing

STT 430/630/ES 760 Lecture Notes: Chapter 6: Hypothesis Testing 1 February 23, 2009 Chapter 6: Introduction to Hypothesis Testing One of the primary uses of statistics is to use data to infer something

### IQ of deaf children example: Are the deaf children lower in IQ? Or are they average? If µ100 and σ 2 225, is the 88.07 from the sample of N59 deaf chi

PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 All inferential statistics have the following in common: Use of some descriptive statistic Use of probability Potential for estimation

### Social Studies 201 Notes for November 19, 2003

1 Social Studies 201 Notes for November 19, 2003 Determining sample size for estimation of a population proportion Section 8.6.2, p. 541. As indicated in the notes for November 17, when sample size is

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm.

Homework #3 is due Friday by 5pm. Homework #4 will be posted to the class website later this week. It will be due Friday, March 7 th, at 5pm. Political Science 15 Lecture 12: Hypothesis Testing Sampling

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### 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

### 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

### TRANSCRIPT: In this lecture, we will talk about both theoretical and applied concepts related to hypothesis testing.

This is Dr. Chumney. The focus of this lecture is hypothesis testing both what it is, how hypothesis tests are used, and how to conduct hypothesis tests. 1 In this lecture, we will talk about both theoretical

### Hypothesis Testing: How to Discriminate between Two Alternatives

CHAPTER 11 Hypothesis Testing: How to Discriminate between Two Alternatives 11.1 What You Will Learn in This Chapter This chapter continues our development of inference to a new situation, one in which

### Lecture 3 : Hypothesis testing and model-fitting

Lecture 3 : Hypothesis testing and model-fitting These dark lectures energy puzzle Lecture 1 : basic descriptive statistics Lecture 2 : searching for correlations Lecture 3 : hypothesis testing and model-fitting

### 8-2 Basics of Hypothesis Testing. Definitions. Rare Event Rule for Inferential Statistics. Null Hypothesis

8-2 Basics of Hypothesis Testing Definitions This section presents individual components of a hypothesis test. We should know and understand the following: How to identify the null hypothesis and alternative

### Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

### Single sample hypothesis testing, II 9.07 3/02/2004

Single sample hypothesis testing, II 9.07 3/02/2004 Outline Very brief review One-tailed vs. two-tailed tests Small sample testing Significance & multiple tests II: Data snooping What do our results mean?

### The Goodness-of-Fit Test

on the Lecture 49 Section 14.3 Hampden-Sydney College Tue, Apr 21, 2009 Outline 1 on the 2 3 on the 4 5 Hypotheses on the (Steps 1 and 2) (1) H 0 : H 1 : H 0 is false. (2) α = 0.05. p 1 = 0.24 p 2 = 0.20

### 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

### 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