Single Sample Means. SOCY601 Alan Neustadtl

Size: px
Start display at page:

Download "Single Sample Means. SOCY601 Alan Neustadtl"

Transcription

1 Single Sample Means SOCY601 Alan Neustadtl

2 The Central Limit Theorem If we have a population measured by a variable with a mean µ and a standard deviation σ, and if all possible random samples of size n are drawn from this population, regardless of the shape of the distribution of the population, then as n becomes large: the distribution of sample means will be approximately normally distributed with a mean equal to the population parameter, and a Standard error equal to the population standard deviation divided by the square-root of N. = µ and s = σ N

3 Important Points We repeatedly take random samples and calculate means. Then we use these means as a variable and create a frequency distribution. This distribution represents the mean of every sample that possibly could be selected. It is a sampling distribution. The distribution of sample means will be normally distributed (particularly if n is large), regardless of the shape of the population. The mean of the sampling distribution is equal to the mean of the population. As sample size increases, the standard error (read standard deviation) of the sampling distribution will decrease.

4 Three Distributions Three Different Types of Distributions Population Sample Sampling Distribution Central Tendency µ = = = = µ N n n Dispersion ( µ ) 2 σ ( ) 2 = N s = n 1 σ σ = n

5 What Does This Mean? Suppose that we have a population with a mean equal to 100 (µ=100) and a standard deviation equal to 15 (σ=15). Assuming that we take a simple random sample of 400 cases (n=400) from this population, we can immediately calculate the standard error of the sampling distribution using the following formula: σ CL CL = = = µ ± ( 1.96)( 0.750) = < µ <

6 The Effect of Sample Size If the sample size was increased to 1,600, the standard error would be smaller and the confidence interval narrower. For example, the standard error would be equal to: σ CL CL = = , 600 = µ ± ( 1.96)( 0.375) = < µ <

7 The Effect of Confidence Size If the sample size is held constant at 1,600, but we used a larger confidence interval, 99% for example, we would see an increase in the range of possible sample means: σ CL CL = = , 600 = µ ± ( 2.58)( 0.375) = < µ <

8 Confidence Intervals ± ( )( σ ) = ± ( ) z z σ n σ x _ µ σ σx µ % Samples x _ µ 1.96σ µ % Samples x σ x µ 2. 58σ + 58σ x µ 2. 99% Samples x

9 Intervals & Level of Confidence Sampling Distribution of the Mean Intervals Extend from Zσ to + Zσ α/2 µ σ x _ 1 - α α/2 = µ Confidence Intervals _ (1 - α) % of Intervals Contain µ. α % Do Not.

10 Important Points All else being equal: As sample size increases, the standard error decreases. As the standard error decreases, the confidence interval decreases. Conversely, small sample sizes are associated with larger standard errors that in turn are associated with larger confidence intervals. Moving from a smaller to larger confidence limit (e.g to 0.99), the confidence interval increases in size it is more inclusive. Conversely, smaller confidence limits (e.g versus 0.99) are associated with smaller confidence intervals they are more exclusive. The smaller the population standard deviation (s), the smaller the standard error and, in turn, the confidence interval. Conversely, the larger the population standard deviation, the larger the standard error and confidence interval.

11 Sample Point Estimates and Confidence Intervals Symbolically a point estimate of a mean is given as. We can place a confidence interval around this value. For example, using a 95% confidence interval (α=0.05) we define boundaries approximately two standard errors below and above the point estimate: ± ( 1.96) σ

12 Sample Point Estimates and Confidence Intervals Similarly, we can construct a 68% confidence interval: ± σ Or a 99% confidence interval: ± ( 2.58) σ

13 Sample Point Estimates and Confidence Intervals In general, confidence intervals can be constructed for any desired level of confidence, 1-α, using this formula: z ± α σ 2

14 Summary of Assumptions We Assume that: 1. the sample for estimatingμ is drawn randomly. 2. we have chosen a sample where n is equal to or greater than that we know σ.

15 Confidence Intervals when the Standard Error is Unknown Typically, we will not know the population parameters. We may be in a position to make assumptions about the mean, but rarely about the standard deviation. We can usually make an estimate of the standard error using the following formula: σ ˆ = s n 1

16 Confidence Intervals when the Standard Error is Unknown When we use this formula, we have to use the t-distribution, not the z-distribution. In general, they are similar. For example, the general formula for confidence intervals becomes: t ± α σˆ = 2 t s ± α 2 n 1

17 z- and t-distributions Similarities to z: There are many t-distributions; their shape varies with the sample size and the sample standard deviation. The t-distribution is bell shaped and has a mean of zero. With large sample sizes (n 150) the t-and z-distributions converge. Difference from z: The use of the t-distribution to test hypotheses assumes that the sample was drawn from a normally distributed population. The use of t is generally robust against the violation of this assumption. A t-distribution for a given sample size has a larger variance than a similar z-distribution. Therefore, the standard error of a t-distribution is larger than that of a similar z-distribution.

18 Student s t Distribution Standard Normal Bell-Shaped Symmetric Fatter Tails t (df = 13) t (df = 5) 0 Z t

19 An Example Using t to Construct Confidence Intervals Research in the 1970s indicated that there was an increase in city size since World War I. But with a reversal in this trend by Using data measuring the percentage change in city populations in 63 American cities, we find that the mean of the difference is with a standard deviation of That is, the point estimate indicated that between 1960 and 1970 there was a decrease in average city size of 1.26%.

20 An Example Using t to Construct Confidence Intervals Using an alpha level of 0.05, there are 62 degrees of freedom (n-l) the tabled value of t is approximately equal to It is approximate because 62 df is not in the table. However, we can use 60 instead. The 95% confidence interval, then, is equal to: CL t ( )( σ ) 95 = ± ˆ 6.32 = 1.26 ± ( 2.00) 63 1 = 1.26 ± = 1.26 ± ( )( ) or: < < 0.33

21 z- and t-tests Besides placing confidence intervals around point estimates of the mean, we can also calculate standard z-tests and t-tests: z µ µ = t = ˆ σ σ

22 An Example of Hypothesis Testing Using Point Estimates If the difference is not equal to zero, do we reject the null hypothesis? To answer that question we need to know what chance or random error can do what kind of differences is chance likely to produce? The central limit theorem provides a distribution based on chance. This allows us to see how chance operates on means.

23 An Example of Hypothesis Testing Using Point Estimates We know that the mean score on an intelligence test in the general population is 100 with a standard deviation of 15. The mean based on a sample of size 100 from a program for accelerated students is 108. Clearly, there is a difference between the population and sample means. What could produce this difference? 1. The program is successful or 2. random error, sampling error, or chance The real question we need to answer is how likely is it that chance produced this difference. Typically, we choose to assume #2 and call it the null hypothesis (H0). In other words, it is not likely that the difference between the sample mean and the population mean is equal exactly to zero; there will generally be some difference. The null hypothesis is the assumption that this difference is due to random error.

24 Hypothesis Testing What could produce differences between observed and expected values? There actually is a difference, or random error, sampling error, or chance. There are five basic steps in hypothesis testing: Assume the null hypothesis of no difference We have to have an idea about the range of outcomes if the null hypothesis is true. We obtain this from an appropriate sampling distribution. We have to decide or set a criterion for enough evidence to be convinced that the null hypothesis is false. This is a significance level called alpha or α. We have to go to the real world and collect data. That is determine some sample statistic. We compare 4 with 3 and reject or fail to reject the null hypothesis. If the value we calculate falls in the critical region or exceeds the critical value associated with α, we must reject the null hypothesis; otherwise we fail to reject it.

25 Null and Alternative Hypotheses First we posit the null H hypothesis: 0 : µ = 0 Next, we choose one of three different alternative hypotheses, depending on a priori expectations: { 1 2-tailed H : µ 0 1-tailed H: µ > 0 1 H : µ < 0 1

26 Hypothesis Testing 1. Assume the null hypothesis of no difference 2. We have to have an idea about the range of outcomes if the null hypothesis is true. We obtain this from an appropriate sampling distribution. 3. We have to decide or set a criterion for enough evidence to be convinced that the null hypothesis is false. This is a significance level called alpha or α. H o : no IQ difference between population and sample H 1 :there is a statistically significant difference in IQ between the population and the sample In this problem, we have a large sample size and we know the population standard deviation. We can safely use the z-distribution to answer this question. It is reasonable to assume that students in an accelerated program should have higher average I.Q. scores. Therefore, we choose to use a onetailed test. Furthermore, since implementing a program like this universally would be expensive we wish to minimize the probability of a Type I error. So, we select α=0.01. In this case, z-critical is equal to

27 Hypothesis Testing 4. We have to go to the real world and collect data. That is determine some sample statistic. 5. We compare 4 with 3 and reject or fail to reject the null hypothesis. If the value we calculate falls in the critical region or exceeds the critical value associated with α, we must reject the null hypothesis; otherwise we fail to reject it. µ z = = 5.33 ˆ σ The calculated z of 5.31 exceeds the z critical of We reject the null hypothesis in favor of the alternative hypothesis knowing that the probability that we have made a Type I error is 1%.

28 Determining How Big A Sample You Need You know that sample size affects the amount of error in parameter estimates ceterus paribus larger samples have less error. This is bound up in the following formula: error t = α 2 ( σ ) ˆ t s n = α 2

29 Determining How Big A Sample You Need So, knowing a and either knowing the population standard deviation or making an estimate of it, you can solve this formula for n, sample size. Consider the following: n t α = 2 error ( σ ) ˆ 2

30 An Example We know that the population mean and standard deviation for the Stanford-Binet intelligence test is 100 and 15 respectively. How large a sample do we need to produce a parameter estimate of the mean within three points of the parameter? Since we know the actual parameters, we can use z.

31 An Example ( 1.96)( 15) 2 n = What if we wanted to reduce the margin of error to one point? How big a sample size do we need to draw? ( 1.96)( 15) 2 n = 1 865

32 Tests Involving Proportions pˆ pˆ pˆ z α ± ( σ ) 2 pˆ = z ( p)( 1 p) α ± = 2 n z ( pˆ)( 1 pˆ) n α ± 2 where: pˆ = n : When is large, ˆ can approximate the value of in the formula for. Note n p p σ p ˆ

33 An Example In a sample of 1,000 American citizens, 637 respond that they trust the president. Using a 95% confidence interval show the range of the population that trusts the president. CL 95 pˆ z ( pˆ)( 1 pˆ) n α = ± 2 = ± 1.96 ( ) = ± < pˆ < ( )( ) 1,000

34 Tests Involving Proportions z = p p s u 1 ( )( ) u n p p u

35 An Example In a sample of 40 students taking an examination, 70% earned a score of 80% or greater. The professor claims success if 80% meet or exceed the goal of mastering 80% of the examination material. Evaluate this examination using a 99% confidence interval. z = 1.58 ( 0.80)( 0.20) 40 z critical is equal to 2.575, so we fail to reject the null hypothesis

36 An Example Using a confidence interval, we get: CL 99 ˆ z ( pˆ)( 1 pˆ) n α = p± 2 = 0.70 ± ( ) = 0.70 ± < pˆ < ( )( ) 40

CIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8

CIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval

More information

Chapter 9 Inferences from Two Samples

Chapter 9 Inferences from Two Samples Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review

More information

HYPOTHESIS TESTING. Hypothesis Testing

HYPOTHESIS TESTING. Hypothesis Testing MBA 605 Business Analytics Don Conant, PhD. HYPOTHESIS TESTING Hypothesis testing involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population.

More information

Last week: Sample, population and sampling distributions finished with estimation & confidence intervals

Last week: Sample, population and sampling distributions finished with estimation & confidence intervals Past weeks: Measures of central tendency (mean, mode, median) Measures of dispersion (standard deviation, variance, range, etc). Working with the normal curve Last week: Sample, population and sampling

More information

Last two weeks: Sample, population and sampling distributions finished with estimation & confidence intervals

Last two weeks: Sample, population and sampling distributions finished with estimation & confidence intervals Past weeks: Measures of central tendency (mean, mode, median) Measures of dispersion (standard deviation, variance, range, etc). Working with the normal curve Last two weeks: Sample, population and sampling

More information

Chapter 5 Confidence Intervals

Chapter 5 Confidence Intervals Chapter 5 Confidence Intervals Confidence Intervals about a Population Mean, σ, Known Abbas Motamedi Tennessee Tech University A point estimate: a single number, calculated from a set of data, that is

More information

STAT Chapter 8: Hypothesis Tests

STAT Chapter 8: Hypothesis Tests STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular

More information

Two-Sample Inferential Statistics

Two-Sample Inferential Statistics The t Test for Two Independent Samples 1 Two-Sample Inferential Statistics In an experiment there are two or more conditions One condition is often called the control condition in which the treatment is

More information

Statistical Inference for Means

Statistical Inference for Means Statistical Inference for Means Jamie Monogan University of Georgia February 18, 2011 Jamie Monogan (UGA) Statistical Inference for Means February 18, 2011 1 / 19 Objectives By the end of this meeting,

More information

STA Module 10 Comparing Two Proportions

STA Module 10 Comparing Two Proportions STA 2023 Module 10 Comparing Two Proportions Learning Objectives Upon completing this module, you should be able to: 1. Perform large-sample inferences (hypothesis test and confidence intervals) to compare

More information

AMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015

AMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015 AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking

More information

PSY 305. Module 3. Page Title. Introduction to Hypothesis Testing Z-tests. Five steps in hypothesis testing

PSY 305. Module 3. Page Title. Introduction to Hypothesis Testing Z-tests. Five steps in hypothesis testing Page Title PSY 305 Module 3 Introduction to Hypothesis Testing Z-tests Five steps in hypothesis testing State the research and null hypothesis Determine characteristics of comparison distribution Five

More information

CENTRAL LIMIT THEOREM (CLT)

CENTRAL LIMIT THEOREM (CLT) CENTRAL LIMIT THEOREM (CLT) A sampling distribution is the probability distribution of the sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic

More information

Chapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc.

Chapter 23. Inferences About Means. Monday, May 6, 13. Copyright 2009 Pearson Education, Inc. Chapter 23 Inferences About Means Sampling Distributions of Means Now that we know how to create confidence intervals and test hypotheses about proportions, we do the same for means. Just as we did before,

More information

T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS

T.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only

More information

Advanced Experimental Design

Advanced Experimental Design Advanced Experimental Design Topic Four Hypothesis testing (z and t tests) & Power Agenda Hypothesis testing Sampling distributions/central limit theorem z test (σ known) One sample z & Confidence intervals

More information

The t-test: A z-score for a sample mean tells us where in the distribution the particular mean lies

The t-test: A z-score for a sample mean tells us where in the distribution the particular mean lies The t-test: So Far: Sampling distribution benefit is that even if the original population is not normal, a sampling distribution based on this population will be normal (for sample size > 30). Benefit

More information

WISE Power Tutorial Answer Sheet

WISE Power Tutorial Answer Sheet ame Date Class WISE Power Tutorial Answer Sheet Power: The B.E.A.. Mnemonic Select true or false for each scenario: (Assuming no other changes) True False 1. As effect size increases, power decreases.

More information

EXAM 3 Math 1342 Elementary Statistics 6-7

EXAM 3 Math 1342 Elementary Statistics 6-7 EXAM 3 Math 1342 Elementary Statistics 6-7 Name Date ********************************************************************************************************************************************** MULTIPLE

More information

Purposes of Data Analysis. Variables and Samples. Parameters and Statistics. Part 1: Probability Distributions

Purposes of Data Analysis. Variables and Samples. Parameters and Statistics. Part 1: Probability Distributions Part 1: Probability Distributions Purposes of Data Analysis True Distributions or Relationships in the Earths System Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution

More information

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă HYPOTHESIS TESTING II TESTS ON MEANS Sorana D. Bolboacă OBJECTIVES Significance value vs p value Parametric vs non parametric tests Tests on means: 1 Dec 14 2 SIGNIFICANCE LEVEL VS. p VALUE Materials and

More information

Mathematical Notation Math Introduction to Applied Statistics

Mathematical Notation Math Introduction to Applied Statistics Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and should be emailed to the instructor

More information

Statistical Inference. Why Use Statistical Inference. Point Estimates. Point Estimates. Greg C Elvers

Statistical Inference. Why Use Statistical Inference. Point Estimates. Point Estimates. Greg C Elvers Statistical Inference Greg C Elvers 1 Why Use Statistical Inference Whenever we collect data, we want our results to be true for the entire population and not just the sample that we used But our sample

More information

Hypothesis Testing. Hypothesis: conjecture, proposition or statement based on published literature, data, or a theory that may or may not be true

Hypothesis Testing. Hypothesis: conjecture, proposition or statement based on published literature, data, or a theory that may or may not be true Hypothesis esting Hypothesis: conjecture, proposition or statement based on published literature, data, or a theory that may or may not be true Statistical Hypothesis: conjecture about a population parameter

More information

Psychology 282 Lecture #4 Outline Inferences in SLR

Psychology 282 Lecture #4 Outline Inferences in SLR Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations

More information

Two Sample Problems. Two sample problems

Two Sample Problems. Two sample problems Two Sample Problems Two sample problems The goal of inference is to compare the responses in two groups. Each group is a sample from a different population. The responses in each group are independent

More information

Inferential statistics

Inferential statistics Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,

More information

CH.9 Tests of Hypotheses for a Single Sample

CH.9 Tests of Hypotheses for a Single Sample CH.9 Tests of Hypotheses for a Single Sample Hypotheses testing Tests on the mean of a normal distributionvariance known Tests on the mean of a normal distributionvariance unknown Tests on the variance

More information

Chapter 7 Comparison of two independent samples

Chapter 7 Comparison of two independent samples Chapter 7 Comparison of two independent samples 7.1 Introduction Population 1 µ σ 1 1 N 1 Sample 1 y s 1 1 n 1 Population µ σ N Sample y s n 1, : population means 1, : population standard deviations N

More information

Practice Questions: Statistics W1111, Fall Solutions

Practice Questions: Statistics W1111, Fall Solutions Practice Questions: Statistics W, Fall 9 Solutions Question.. The standard deviation of Z is 89... P(=6) =..3. is definitely inside of a 95% confidence interval for..4. (a) YES (b) YES (c) NO (d) NO Questions

More information

Chapter 23. Inference About Means

Chapter 23. Inference About Means Chapter 23 Inference About Means 1 /57 Homework p554 2, 4, 9, 10, 13, 15, 17, 33, 34 2 /57 Objective Students test null and alternate hypotheses about a population mean. 3 /57 Here We Go Again Now that

More information

1 Descriptive statistics. 2 Scores and probability distributions. 3 Hypothesis testing and one-sample t-test. 4 More on t-tests

1 Descriptive statistics. 2 Scores and probability distributions. 3 Hypothesis testing and one-sample t-test. 4 More on t-tests Overall Overview INFOWO Statistics lecture S3: Hypothesis testing Peter de Waal Department of Information and Computing Sciences Faculty of Science, Universiteit Utrecht 1 Descriptive statistics 2 Scores

More information

Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.

Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6. Chapter 7 Reading 7.1, 7.2 Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.112 Introduction In Chapter 5 and 6, we emphasized

More information

Lecture on Null Hypothesis Testing & Temporal Correlation

Lecture on Null Hypothesis Testing & Temporal Correlation Lecture on Null Hypothesis Testing & Temporal Correlation CS 590.21 Analysis and Modeling of Brain Networks Department of Computer Science University of Crete Acknowledgement Resources used in the slides

More information

Section 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples

Section 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means

More information

Statistics Primer. ORC Staff: Jayme Palka Peter Boedeker Marcus Fagan Trey Dejong

Statistics Primer. ORC Staff: Jayme Palka Peter Boedeker Marcus Fagan Trey Dejong Statistics Primer ORC Staff: Jayme Palka Peter Boedeker Marcus Fagan Trey Dejong 1 Quick Overview of Statistics 2 Descriptive vs. Inferential Statistics Descriptive Statistics: summarize and describe data

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 03 The Chi-Square Distributions Dr. Neal, Spring 009 The chi-square distributions can be used in statistics to analyze the standard deviation of a normally distributed measurement and to test the

More information

The t-statistic. Student s t Test

The t-statistic. Student s t Test The t-statistic 1 Student s t Test When the population standard deviation is not known, you cannot use a z score hypothesis test Use Student s t test instead Student s t, or t test is, conceptually, very

More information

y = a + bx 12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation Review: Interpreting Computer Regression Output

y = a + bx 12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation Review: Interpreting Computer Regression Output 12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation y = a + bx y = dependent variable a = intercept b = slope x = independent variable Section 12.1 Inference for Linear

More information

Introduction to Business Statistics QM 220 Chapter 12

Introduction to Business Statistics QM 220 Chapter 12 Department of Quantitative Methods & Information Systems Introduction to Business Statistics QM 220 Chapter 12 Dr. Mohammad Zainal 12.1 The F distribution We already covered this topic in Ch. 10 QM-220,

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation σ of a normally distributed measurement and to test the goodness

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Part 1 Sampling Distributions, Point Estimates & Confidence Intervals Inferential statistics are used to draw inferences (make conclusions/judgements) about a population from a sample.

More information

Lab #12: Exam 3 Review Key

Lab #12: Exam 3 Review Key Psychological Statistics Practice Lab#1 Dr. M. Plonsky Page 1 of 7 Lab #1: Exam 3 Review Key 1) a. Probability - Refers to the likelihood that an event will occur. Ranges from 0 to 1. b. Sampling Distribution

More information

Harvard University. Rigorous Research in Engineering Education

Harvard University. Rigorous Research in Engineering Education Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected

More information

Business Statistics. Lecture 10: Course Review

Business Statistics. Lecture 10: Course Review Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,

More information

Problem Set 4 - Solutions

Problem Set 4 - Solutions Problem Set 4 - Solutions Econ-310, Spring 004 8. a. If we wish to test the research hypothesis that the mean GHQ score for all unemployed men exceeds 10, we test: H 0 : µ 10 H a : µ > 10 This is a one-tailed

More information

Lecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series

Lecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 7 Estimates and Sample Sizes 7-1 Overview 7-2 Estimating a Population Proportion 7-3

More information

PSY 216. Assignment 9 Answers. Under what circumstances is a t statistic used instead of a z-score for a hypothesis test

PSY 216. Assignment 9 Answers. Under what circumstances is a t statistic used instead of a z-score for a hypothesis test PSY 216 Assignment 9 Answers 1. Problem 1 from the text Under what circumstances is a t statistic used instead of a z-score for a hypothesis test The t statistic should be used when the population standard

More information

Sampling, Confidence Interval and Hypothesis Testing

Sampling, Confidence Interval and Hypothesis Testing Sampling, Confidence Interval and Hypothesis Testing Christopher Grigoriou Executive MBA HEC Lausanne 2007-2008 1 Sampling : Careful with convenience samples! World War II: A statistical study to decide

More information

Quantitative Methods for Economics, Finance and Management (A86050 F86050)

Quantitative Methods for Economics, Finance and Management (A86050 F86050) Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge

More information

CHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities:

CHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: CHAPTER 9, 10 Hypothesis Testing Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: The person is guilty. The person is innocent. To

More information

Difference in two or more average scores in different groups

Difference in two or more average scores in different groups ANOVAs Analysis of Variance (ANOVA) Difference in two or more average scores in different groups Each participant tested once Same outcome tested in each group Simplest is one-way ANOVA (one variable as

More information

Exam 2 (KEY) July 20, 2009

Exam 2 (KEY) July 20, 2009 STAT 2300 Business Statistics/Summer 2009, Section 002 Exam 2 (KEY) July 20, 2009 Name: USU A#: Score: /225 Directions: This exam consists of six (6) questions, assessing material learned within Modules

More information

Statistics 251: Statistical Methods

Statistics 251: Statistical Methods Statistics 251: Statistical Methods 1-sample Hypothesis Tests Module 9 2018 Introduction We have learned about estimating parameters by point estimation and interval estimation (specifically confidence

More information

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007) FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter

More information

Multiple t Tests. Introduction to Analysis of Variance. Experiments with More than 2 Conditions

Multiple t Tests. Introduction to Analysis of Variance. Experiments with More than 2 Conditions Introduction to Analysis of Variance 1 Experiments with More than 2 Conditions Often the research that psychologists perform has more conditions than just the control and experimental conditions You might

More information

Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing

Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Business Statistics: Lecture 8: Introduction to Estimation & Hypothesis Testing Agenda Introduction to Estimation Point estimation Interval estimation Introduction to Hypothesis Testing Concepts en terminology

More information

1 MA421 Introduction. Ashis Gangopadhyay. Department of Mathematics and Statistics. Boston University. c Ashis Gangopadhyay

1 MA421 Introduction. Ashis Gangopadhyay. Department of Mathematics and Statistics. Boston University. c Ashis Gangopadhyay 1 MA421 Introduction Ashis Gangopadhyay Department of Mathematics and Statistics Boston University c Ashis Gangopadhyay 1.1 Introduction 1.1.1 Some key statistical concepts 1. Statistics: Art of data analysis,

More information

10/4/2013. Hypothesis Testing & z-test. Hypothesis Testing. Hypothesis Testing

10/4/2013. Hypothesis Testing & z-test. Hypothesis Testing. Hypothesis Testing & z-test Lecture Set 11 We have a coin and are trying to determine if it is biased or unbiased What should we assume? Why? Flip coin n = 100 times E(Heads) = 50 Why? Assume we count 53 Heads... What could

More information

STAT 515 fa 2016 Lec Statistical inference - hypothesis testing

STAT 515 fa 2016 Lec Statistical inference - hypothesis testing STAT 515 fa 2016 Lec 20-21 Statistical inference - hypothesis testing Karl B. Gregory Wednesday, Oct 12th Contents 1 Statistical inference 1 1.1 Forms of the null and alternate hypothesis for µ and p....................

More information

Chapter 24. Comparing Means

Chapter 24. Comparing Means Chapter 4 Comparing Means!1 /34 Homework p579, 5, 7, 8, 10, 11, 17, 31, 3! /34 !3 /34 Objective Students test null and alternate hypothesis about two!4 /34 Plot the Data The intuitive display for comparing

More information

Stat 529 (Winter 2011) Experimental Design for the Two-Sample Problem. Motivation: Designing a new silver coins experiment

Stat 529 (Winter 2011) Experimental Design for the Two-Sample Problem. Motivation: Designing a new silver coins experiment Stat 529 (Winter 2011) Experimental Design for the Two-Sample Problem Reading: 2.4 2.6. Motivation: Designing a new silver coins experiment Sample size calculations Margin of error for the pooled two sample

More information

Hypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals

Hypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals Hypothesis Testing and Confidence Intervals (Part 2): Cohen s d, Logic of Testing, and Confidence Intervals Lecture 9 Justin Kern April 9, 2018 Measuring Effect Size: Cohen s d Simply finding whether a

More information

Chapter 7: Hypothesis Testing

Chapter 7: Hypothesis Testing Chapter 7: Hypothesis Testing *Mathematical statistics with applications; Elsevier Academic Press, 2009 The elements of a statistical hypothesis 1. The null hypothesis, denoted by H 0, is usually the nullification

More information

Chapter 7: Hypothesis Testing - Solutions

Chapter 7: Hypothesis Testing - Solutions Chapter 7: Hypothesis Testing - Solutions 7.1 Introduction to Hypothesis Testing The problem with applying the techniques learned in Chapter 5 is that typically, the population mean (µ) and standard deviation

More information

Introduction to the Analysis of Variance (ANOVA) Computing One-Way Independent Measures (Between Subjects) ANOVAs

Introduction to the Analysis of Variance (ANOVA) Computing One-Way Independent Measures (Between Subjects) ANOVAs Introduction to the Analysis of Variance (ANOVA) Computing One-Way Independent Measures (Between Subjects) ANOVAs The Analysis of Variance (ANOVA) The analysis of variance (ANOVA) is a statistical technique

More information

KDF2C QUANTITATIVE TECHNIQUES FOR BUSINESSDECISION. Unit : I - V

KDF2C QUANTITATIVE TECHNIQUES FOR BUSINESSDECISION. Unit : I - V KDF2C QUANTITATIVE TECHNIQUES FOR BUSINESSDECISION Unit : I - V Unit I: Syllabus Probability and its types Theorems on Probability Law Decision Theory Decision Environment Decision Process Decision tree

More information

ME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV

ME3620. Theory of Engineering Experimentation. Spring Chapter IV. Decision Making for a Single Sample. Chapter IV Theory of Engineering Experimentation Chapter IV. Decision Making for a Single Sample Chapter IV 1 4 1 Statistical Inference The field of statistical inference consists of those methods used to make decisions

More information

Soc3811 Second Midterm Exam

Soc3811 Second Midterm Exam Soc38 Second Midterm Exam SEMI-OPE OTE: One sheet of paper, signed & turned in with exam booklet Bring our Own Pencil with Eraser and a Hand Calculator! Standardized Scores & Probability If we know the

More information

Sampling Distributions: Central Limit Theorem

Sampling Distributions: Central Limit Theorem Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)

More information

A proportion is the fraction of individuals having a particular attribute. Can range from 0 to 1!

A proportion is the fraction of individuals having a particular attribute. Can range from 0 to 1! Proportions A proportion is the fraction of individuals having a particular attribute. It is also the probability that an individual randomly sampled from the population will have that attribute Can range

More information

Statistics for IT Managers

Statistics for IT Managers Statistics for IT Managers 95-796, Fall 2012 Module 2: Hypothesis Testing and Statistical Inference (5 lectures) Reading: Statistics for Business and Economics, Ch. 5-7 Confidence intervals Given the sample

More information

Basics of Experimental Design. Review of Statistics. Basic Study. Experimental Design. When an Experiment is Not Possible. Studying Relations

Basics of Experimental Design. Review of Statistics. Basic Study. Experimental Design. When an Experiment is Not Possible. Studying Relations Basics of Experimental Design Review of Statistics And Experimental Design Scientists study relation between variables In the context of experiments these variables are called independent and dependent

More information

Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test

Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test 1 Section 10.1 (Part 2 of 2) Significance Tests: Power of a Test Learning Objectives After this section, you should be able to DESCRIBE the relationship between the significance level of a test, P(Type

More information

Chapter 12: Inference about One Population

Chapter 12: Inference about One Population Chapter 1: Inference about One Population 1.1 Introduction In this chapter, we presented the statistical inference methods used when the problem objective is to describe a single population. Sections 1.

More information

ANOVA TESTING 4STEPS. 1. State the hypothesis. : H 0 : µ 1 =

ANOVA TESTING 4STEPS. 1. State the hypothesis. : H 0 : µ 1 = Introduction to Statistics in Psychology PSY 201 Professor Greg Francis Lecture 35 ANalysis Of VAriance Ignoring (some) variability TESTING 4STEPS 1. State the hypothesis. : H 0 : µ 1 = µ 2 =... = µ K,

More information

Review. One-way ANOVA, I. What s coming up. Multiple comparisons

Review. One-way ANOVA, I. What s coming up. Multiple comparisons Review One-way ANOVA, I 9.07 /15/00 Earlier in this class, we talked about twosample z- and t-tests for the difference between two conditions of an independent variable Does a trial drug work better than

More information

Statistics 301: Probability and Statistics 1-sample Hypothesis Tests Module

Statistics 301: Probability and Statistics 1-sample Hypothesis Tests Module Statistics 301: Probability and Statistics 1-sample Hypothesis Tests Module 9 2018 Student s t graphs For the heck of it: x

More information

1/24/2008. Review of Statistical Inference. C.1 A Sample of Data. C.2 An Econometric Model. C.4 Estimating the Population Variance and Other Moments

1/24/2008. Review of Statistical Inference. C.1 A Sample of Data. C.2 An Econometric Model. C.4 Estimating the Population Variance and Other Moments /4/008 Review of Statistical Inference Prepared by Vera Tabakova, East Carolina University C. A Sample of Data C. An Econometric Model C.3 Estimating the Mean of a Population C.4 Estimating the Population

More information

23. MORE HYPOTHESIS TESTING

23. MORE HYPOTHESIS TESTING 23. MORE HYPOTHESIS TESTING The Logic Behind Hypothesis Testing For simplicity, consider testing H 0 : µ = µ 0 against the two-sided alternative H A : µ µ 0. Even if H 0 is true (so that the expectation

More information

Hypothesis Tests and Estimation for Population Variances. Copyright 2014 Pearson Education, Inc.

Hypothesis Tests and Estimation for Population Variances. Copyright 2014 Pearson Education, Inc. Hypothesis Tests and Estimation for Population Variances 11-1 Learning Outcomes Outcome 1. Formulate and carry out hypothesis tests for a single population variance. Outcome 2. Develop and interpret confidence

More information

Sampling distribution of t. 2. Sampling distribution of t. 3. Example: Gas mileage investigation. II. Inferential Statistics (8) t =

Sampling distribution of t. 2. Sampling distribution of t. 3. Example: Gas mileage investigation. II. Inferential Statistics (8) t = 2. The distribution of t values that would be obtained if a value of t were calculated for each sample mean for all possible random of a given size from a population _ t ratio: (X - µ hyp ) t s x The result

More information

16.400/453J Human Factors Engineering. Design of Experiments II

16.400/453J Human Factors Engineering. Design of Experiments II J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential

More information

AP Statistics Ch 12 Inference for Proportions

AP Statistics Ch 12 Inference for Proportions Ch 12.1 Inference for a Population Proportion Conditions for Inference The statistic that estimates the parameter p (population proportion) is the sample proportion p ˆ. p ˆ = Count of successes in the

More information

Answer keys for Assignment 10: Measurement of study variables (The correct answer is underlined in bold text)

Answer keys for Assignment 10: Measurement of study variables (The correct answer is underlined in bold text) Answer keys for Assignment 10: Measurement of study variables (The correct answer is underlined in bold text) 1. A quick and easy indicator of dispersion is a. Arithmetic mean b. Variance c. Standard deviation

More information

Making Inferences About Parameters

Making Inferences About Parameters Making Inferences About Parameters Parametric statistical inference may take the form of: 1. Estimation: on the basis of sample data we estimate the value of some parameter of the population from which

More information

Visual interpretation with normal approximation

Visual interpretation with normal approximation Visual interpretation with normal approximation H 0 is true: H 1 is true: p =0.06 25 33 Reject H 0 α =0.05 (Type I error rate) Fail to reject H 0 β =0.6468 (Type II error rate) 30 Accept H 1 Visual interpretation

More information

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.

Review: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses. 1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately

More information

POLI 443 Applied Political Research

POLI 443 Applied Political Research POLI 443 Applied Political Research Session 4 Tests of Hypotheses The Normal Curve Lecturer: Prof. A. Essuman-Johnson, Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College

More information

7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between

7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between 7.2 One-Sample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation

More information

Chapter. Hypothesis Testing with Two Samples. Copyright 2015, 2012, and 2009 Pearson Education, Inc. 1

Chapter. Hypothesis Testing with Two Samples. Copyright 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter 8 Hypothesis Testing with Two Samples Copyright 2015, 2012, and 2009 Pearson Education, Inc 1 Two Sample Hypothesis Test Compares two parameters from two populations Sampling methods: Independent

More information

How do we compare the relative performance among competing models?

How do we compare the relative performance among competing models? How do we compare the relative performance among competing models? 1 Comparing Data Mining Methods Frequent problem: we want to know which of the two learning techniques is better How to reliably say Model

More information

Section 6.2 Hypothesis Testing

Section 6.2 Hypothesis Testing Section 6.2 Hypothesis Testing GIVEN: an unknown parameter, and two mutually exclusive statements H 0 and H 1 about. The Statistician must decide either to accept H 0 or to accept H 1. This kind of problem

More information

Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1

Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1 Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences

More information

An inferential procedure to use sample data to understand a population Procedures

An inferential procedure to use sample data to understand a population Procedures Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors

More information

A3. Statistical Inference

A3. Statistical Inference Appendi / A3. Statistical Inference / Mean, One Sample-1 A3. Statistical Inference Population Mean μ of a Random Variable with known standard deviation σ, and random sample of size n 1 Before selecting

More information

" M A #M B. Standard deviation of the population (Greek lowercase letter sigma) σ 2

 M A #M B. Standard deviation of the population (Greek lowercase letter sigma) σ 2 Notation and Equations for Final Exam Symbol Definition X The variable we measure in a scientific study n The size of the sample N The size of the population M The mean of the sample µ The mean of the

More information

hypothesis a claim about the value of some parameter (like p)

hypothesis a claim about the value of some parameter (like p) Testing hypotheses hypothesis a claim about the value of some parameter (like p) significance test procedure to assess the strength of evidence provided by a sample of data against the claim of a hypothesized

More information

Outline. PubH 5450 Biostatistics I Prof. Carlin. Confidence Interval for the Mean. Part I. Reviews

Outline. PubH 5450 Biostatistics I Prof. Carlin. Confidence Interval for the Mean. Part I. Reviews Outline Outline PubH 5450 Biostatistics I Prof. Carlin Lecture 11 Confidence Interval for the Mean Known σ (population standard deviation): Part I Reviews σ x ± z 1 α/2 n Small n, normal population. Large

More information

LECTURE 12 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING

LECTURE 12 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING LECTURE 1 CONFIDENCE INTERVAL AND HYPOTHESIS TESTING INTERVAL ESTIMATION Point estimation of : The inference is a guess of a single value as the value of. No accuracy associated with it. Interval estimation

More information