# Sociology 6Z03 Topic 15: Statistical Inference for Means

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Outline: Statistical Inference for Means Introduction Comparing Two Means From Independent Samples John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

2 Introduction We have learned how to perform classical statistical inference constructing confidence intervals and testing hypotheses for a population mean µ when (unrealistically) the population standard deviation σ is known. Our procedures made use of the fact that the sample mean has the approximate sampling distribution N(µ, σ/ n). Almost regardless of the population distribution of X, this approximation grows more accurate as the sample size n grows (the central limit theorem); and if the population distribution of X is normal i.e., X N(µ, σ) then the result is exact, even for small samples. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Introduction In this lecture, we will learn how to perform statistical inference for a population mean µ when the population standard deviation σ is not known. We will also learn how to construct confidence intervals and perform hypothesis tests for the difference between two means from independently sampled populations a common research situation. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

3 Assumptions The procedures that we are about to develop depend upon two key assumptions: 1 The data are a simple random sample from a much larger population. 2 The distribution of the variable in the population is a normal distribution with mean µ and standard deviation σ, both of which are unknown. If the distribution is single-peaked, roughly symmetric, and doesn t have very heavy tails (which tend to give rise to outliers), the assumption of normality isn t critical unless the sample size is very small. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 The Standard Error of the Sample Mean Although sample means are normally distributed with mean µ and standard deviation σ/ n, we cannot make direct use of this fact when σ is not known. We can, however, estimate the standard deviation of x by using the sample standard deviation s in place of the unknown σ. The resulting estimated standard deviation of x is called the standard error of x: SE(x) = s n A note on terminology: Some authors refer to SD(x) = σ/ n as the standard error of x ; then s/ n is called the estimated standard error of x. Following Moore, I will reserve the term standard error for the estimate that is, s/ n. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

4 The t Distributions When the population standard deviation σ is known, tests and confidence intervals for µ are based on the standardized sample mean z = x µ σ/ n where z N(0, 1). When σ is not known, we can calculate the analogous statistic, t = x µ s/ n but this statistic is not normally distributed. Instead, if the population distribution of X is a normal distribution, then this new statistic follows Student s t-distribution with n 1 degrees of freedom. Student was the nom-de-plume of the discoverer of the t-distribution W. S. Gosset, a statistician who worked for Guinness brewery around the turn of the 20th century. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 The t Distributions There is a different t-distribution for each number of degrees of freedom, 1, 2,.... The degrees of freedom for t come from the denominator of the sample standard deviation s, (x s = i x) 2 n 1 As the degrees of freedom grow, the t-distribution approaches the normal distribution. For small degrees of freedom, the t-distribution is more spread out than the normal distribution, reflecting the additional uncertainty that results from having to estimate σ rather than knowing it exactly. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

5 Some t Distributions Density N(0,1) t-distributions: N(0, 1) = t( ). t(10) t(2) John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests Confidence intervals and tests based on the t-distribution are very similar to intervals and tests based on the normal distribution. Rather than using critical values from the normal distribution, however, we need to use critical values of t. These values may be found for various degrees of freedom in Table C of the text. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

6 t Confidence Intervals and Hypothesis Tests: Example Consider the following (familiar) example: An educational researcher wants to know whether a new method of teaching statistics is superior to the old method. Ten instructors who each teach two sections of an introductory statistics class are recruited into the study. Each instructor has one of his or her sections assigned at random to the new teaching method; the other section is taught by the old method. At the end of the study, the students in all sections of the course take a common exam. The average grade on the exam in each section is contained in the table on the next slide, along with the difference for each instructor between the scores for the sections taught according to the new and old methods. We previously analyzed these data without the proper tools, by simply pretending that the population standard deviation σ is the same as the sample standard deviation s. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests: Example Instructor New Method Old Method Difference Class Mean Class Mean x i The mean difference is x = 9.40, and the standard deviation of the n = 10 differences is s = John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

7 t Confidence Interval: Example Using the t distribution to construct a 95-percent confidence interval: Degrees of freedom = n 1 = 10 1 = 9. The critical value of t with 9 degrees of freedom for the C =.95 level of confidence has the area.025 to the right; from Table C, this value is t = t* 0 t* = t with 9 d.f. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Interval: Example Thought Question TRUE or FALSE: t = is larger than the corresponding critical value of z (which is z = 1.960), and therefore t produces a narrower, more precise confidence interval than when the population standard deviation σ is known. A TRUE B FALSE C I don t know. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

8 t Confidence Interval: Example The 95-percent confidence interval is x ± t s = 9.40 ± n 10 = 9.40 ± 5.99 = 3.41 to points John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example Alternatively, to test the null hypothesis that the new method is no better than the old H 0 : µ = 0 against the alternative hypothesis that the new method is better we calculate the test statistic H a : µ > 0 t = x µ 0 s/ n = / 10 = If the null hypothesis is true, then this test statistic follows a t distribution with n 1 = 10 1 = 9 degrees of freedom. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

9 t Hypothesis Test: Example Most computer programs report the exact P-value for a t-test, but if we need to use the t-table we won t usually be able to find a precise P-value. Because the alternative hypothesis is directional, and specifies a positive value of µ, we find the P-value by looking in the right-hand tail of the t-distribution with 9 degrees of freedom: One-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example Thought Question The observed value of t is What is the P-value for the hypothesis test? A.005 < P <.0025 B.0025 < P <.005 C P =.001 D I don t know. One-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

10 t Hypothesis Test: Example P t with 9 d.f. 0 t* = prob. to the right =.005 observed t = Finding the P-value. t* = prob. to the right =.0025 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example, Two-Sided Test Thought Question TRUE or FALSE: If the alternative hypothesis were nondirectional, H a : µ = 0, then, in comparison with the previous one-sided test, we would double the P-value, which would be reported as t = 3.547, df = 9,.005 < P <.01, two-tail. A TRUE B FALSE C I don t know. One-Sided P Two-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

11 t Confidence Intervals and Hypothesis Tests: Caution A practical limitation of t-tests and t-intervals is that in small samples they depend upon the assumption that the population is normally distributed. In large samples, t-tests and t-intervals are generally quite accurate even if the population is not normal, but in large samples inferences based on the t-distribution and the normal distribution are essentially indistinguishable. We should check the assumption of normality by examining the distribution of the data, but in small samples where the assumption really counts it is hard to assess departures from normality. We should, however, be on the lookout for outliers, more than one mode, and serious skewness. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests: Caution A stem-plot for the illustrative dataset: There s nothing obviously problematic here: The distribution is single-peaked, roughly symmetric, and has no outliers. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

12 t Confidence Intervals and Hypothesis Tests: Matched Pairs A very common use of t-intervals and t-tests for a single mean is for matched-pairs data. The illustration above is an example of matched pairs: Each instructor taught two classes, one of which was taught according to the old method, and the other according to the new method. The design of the study would be fundamentally different if the 20 classes were taught by 20 different instructors. In this case, 10 instructors could be randomly assigned to teach by the new method, 10 by the old method. This alternative study uses two independent samples rather than matched pairs. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Confidence Intervals and Hypothesis Tests: Matched Pairs Some other common examples of matched pairs: Before-After Studies: We examine the average auto accident rate x in n = 10 jurisdictions before and after the imposition of random spot checks. Note that this is an inherently weak design, because we do not control for what would have happened had the spot checks not started. A comparative experiment, in which some jurisdictions have spot checks imposed and others not, would be a better design. Sampling of Natural Pairs: We sample n = 100 heterosexual married couples and calculate the average difference x between husbands and wives incomes. This is different from sampling husbands and wives (or men and women) independently. Where they are appropriate, matched-pairs designs tend to be more powerful than independent-samples designs, because each pair serves as its own control. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

13 Comparing Two Means From Independent Samples As mentioned, comparing two means from independent samples is a very common research situation. We will proceed under the following assumptions: 1 We have two independent simple random samples from two different populations. Matched or paired samples are examples of dependent samples. Often a simple random sample of a general population is divided into two independent subsamples. For example, a general simple random sample of the adult Canadian population can be divided into independent subsamples of men and women. 2 Each population is normally distributed, but with unknown means and standard deviations. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Notation and Null Hypothesis We will use the following notation to describe the populations: Population Variable Mean Standard Deviation 1 x 1 µ 1 σ 1 2 x 2 µ 2 σ 2 Our interest is in comparing the population means µ 1 and µ 2. We can do this either by constructing a confidence interval for the difference µ 1 µ 2 or by testing the null hypothesis of equal population means, H 0 : µ 1 = µ 2 which is equivalent to the null hypothesis of no difference in population means, H 0 : µ 1 µ 2 = 0 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

14 Comparing Two Means From Independent Samples Sample Data We can lay out our sample data as follows: Population Sample Size Sample Mean Sample Std. Dev. 1 n 1 x 1 s 1 2 n 2 x 2 s 2 It is natural to use the sample difference x 1 x 2 to estimate the population difference µ 1 µ 2. Because the population standard deviations σ 1 and σ 2 are unknown, we will use the corresponding sample standard deviations s 1 and s 2 to estimate them. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Sampling Distribution of the Difference in Sample Means To perform statistical inference for the difference µ 1 µ 2 based upon x 1 x 2, we need to know the properties of the sampling distribution of x 1 x 2 : The mean of x 1 x 2 is µ 1 µ 2. The variance of x 1 x 2 is so the standard deviation of x 1 x 2 is σ 2 1 n 1 + σ2 2 n 2 SD(x 1 x 2 ) = σ 2 1 n 1 + σ2 2 n 2 If the two populations are normally distributed then so is the difference in sample means, x 1 x 2. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

15 Comparing Two Means From Independent Samples Sampling Distribution of the Difference in Sample Means Thought Question TRUE or FALSE: The difference in sample means x 1 x 2 is an unbiased estimator of the difference in population means µ 1 µ 2. A TRUE B FALSE C I don t know. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Two-Sample t-tests and t-intervals If the population standard deviations σ 1 and σ 2 were known, then we could base statistical inference on the normal distribution, because the standardized value z = (x 1 x 2 ) (µ 1 µ 2 ) σ1 2 + σ2 2 n 1 n 2 follows the standard normal distribution, N(0, 1). John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

16 Comparing Two Means From Independent Samples Two-Sample t-tests and t-intervals Instead, we use the two-sample t-statistic t = (x 1 x 2 ) (µ 1 µ 2 ) s1 2 + s2 2 n 1 n 2 Under the assumptions of simple random sampling and normal populations this statistic follows an approximate t-distribution. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Two-Sample t Confidence Interval To construct a level-c confidence interval, calculate (x 1 x 2 ) ± t s1 2 + s2 2 n 1 n 2 where t is the appropriate critical value from the t-distribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1 (or estimated by a complex formula, called the Welch-Satterthwaite equation, given in the text). John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

17 Comparing Two Means From Independent Samples Two-Sample t Hypothesis Test To test the null hypothesis H 0 : µ 1 = µ 2, calculate the statistic t = x 1 x 2 s1 2 + s2 2 n 1 n 2 and refer this statistic to the t-distribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1 (or estimated by the Welch-Satterthwaite formula). Notice that the numerator of this test statistic comes from (x 1 x 2 ) 0; That is, 0 is the hypothesized value of µ 1 µ 2. We perform a one-sided or two-sided test depending upon whether the alternative hypothesis is directional, or nondirectional, H a : µ 1 = µ 2. H a : µ 1 > µ 2 or H a : µ 1 < µ 2, John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example To illustrate the two-sample t procedures, I ve employed data drawn from a survey of sociology students at McMaster, dividing the students who responded to the survey into two groups: (1) Those with grade-point averages of B or less; and (2) those with grade-point averages of B+ or better. In each group, I ve calculated the mean and standard deviation of number of hours of TV viewing per week. The results are as follows: Grade-Point Average n x s B or lower B+ or higher John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

18 Comparing Two Means From Independent Samples Example: 95 Percent Confidence Interval The smaller of n 1 1 and n 2 1 is 45 1 = 44. The critical value of t with 40 d.f. the closest value below 44 d.f. in the t table is t = The 95 percent confidence interval is therefore (x 1 x 2 ) ± t s1 2 + s2 2 n 1 n = ( ) ± = 4.57 ± 3.67 = 0.90 to 8.24 hours John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example: Hypothesis Test To test H 0 : µ 1 = µ 2 against the one-sided alternative hypothesis H a : µ 1 > µ 2 (students with lower grades watch more TV), calculate t = x 1 x 2 s1 2 + s2 2 n 1 n 2 = = John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

19 Comparing Two Means From Independent Samples Example: Hypothesis Test Thought Question Entering the t table with 40 d.f. for our test statistic t = 2.514, what is the P-value for the test? A.025 < P <.05. B.01 < P <.02. C.005 < P <.01. D I don t know. One-Sided P Two-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example: Hypothesis Test Thought Question If we specified a two-sided alternative hypothesis, what would be the P-value associated with our test statistic t = 2.514? A.005 < P <.01. B.01 < P <.02. C.0025 < P <.005. D I don t know. One-Sided P Two-Sided P t John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

20 TV Hours/Week Comparing Two Means From Independent Samples Example: Checking the Data Examining the data graphically reveals problems: The distribution of hours of TV watching within groups is somewhat positively skewed and there are outliers in both groups (with the dots representing the group means): B or lower B+ or higher Grade Point Average John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples Example: Checking the Data With samples as large as 95 and 45, the t test and t interval are approximately correct even when the distributions are quite skewed (and hence non-normal). We say that the validity of the t procedures is robust with respect to departures from the assumption of normality. We might wonder, however, whether it would be better to use the group medians, rather than the group means, to summarize the data. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

21 Comparing Two Means From Independent Samples Example: Checking the Data As well, eliminating the outliers changes the results somewhat: Grade-Point Average n x s B or lower B+ or higher t = 5.19 df 42 1 = 41 P <.0005 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41

### 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

### 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

### 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

### Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

### 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

### 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

### 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

### 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.

### 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,

### 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

### 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

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### 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

### Selected Nonparametric and Parametric Statistical Tests for Two-Sample Cases 1

Selected Nonparametric and Parametric Statistical Tests for Two-Sample Cases The T-statistic is used to test differences in the means of two groups. The grouping variable is categorical and data for the

### 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

### 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

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### 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

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### 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

### Mind on Statistics. Chapter 13

Mind on Statistics Chapter 13 Sections 13.1-13.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question

### 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 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

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### 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,

### 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

### Nonparametric tests, Bootstrapping

Nonparametric tests, Bootstrapping http://www.isrec.isb-sib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis

### Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide

### Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)

Confidence Interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated

### 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

### 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

### STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico. Fall 2013

STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico Fall 2013 CHAPTER 18 INFERENCE ABOUT A POPULATION MEAN. Conditions for Inference about mean

### Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is

### 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

### Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

### 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

### 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

### Practice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35

Practice Exam Use the following to answer questions 1-2: A census is done in a given region. Following are the populations of the towns in that particular region (in thousands): 35, 46, 52, 63, 64, 71,

### 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

### 3. Nonparametric methods

3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests

### 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

### The Wilcoxon Rank-Sum Test

1 The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the twosample t-test which is based solely on the order in which the observations from the two samples fall. We

### Part 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217

Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing

### AP * Statistics Review

AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this

### Histogram. Graphs, and measures of central tendency and spread. Alternative: density (or relative frequency ) plot /13/2004

Graphs, and measures of central tendency and spread 9.07 9/13/004 Histogram If discrete or categorical, bars don t touch. If continuous, can touch, should if there are lots of bins. Sum of bin heights

### 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

### 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

### MTH 140 Statistics Videos

MTH 140 Statistics Videos Chapter 1 Picturing Distributions with Graphs Individuals and Variables Categorical Variables: Pie Charts and Bar Graphs Categorical Variables: Pie Charts and Bar Graphs Quantitative

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### Introduction to Stata

Introduction to Stata September 23, 2014 Stata is one of a few statistical analysis programs that social scientists use. Stata is in the mid-range of how easy it is to use. Other options include SPSS,

### Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance

2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

### 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

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Opgaven Onderzoeksmethoden, Onderdeel Statistiek

Opgaven Onderzoeksmethoden, Onderdeel Statistiek 1. What is the measurement scale of the following variables? a Shoe size b Religion c Car brand d Score in a tennis game e Number of work hours per week

### Hypothesis Testing. Bluman Chapter 8

CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

### Analysis of numerical data S4

Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak k.jozwiak@nki.nl 3rd November 2015 1/42 Hypothesis tests: numerical and ordinal data 1 group:

### 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

### An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.

### 9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1 - ȳ 2 ) = SE d - = s d n d

CHAPTER 9 Comparison of Paired Samples 9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1 - ȳ 2 ) = SE d - = s d n d =.68 4 =.34. (b) H 0 : The mean

### 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

### Outline. Correlation & Regression, III. Review. Relationship between r and regression

Outline Correlation & Regression, III 9.07 4/6/004 Relationship between correlation and regression, along with notes on the correlation coefficient Effect size, and the meaning of r Other kinds of correlation

### 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

### Suppose we want to compare the average effectiveness of two treatments in a completely randomized experiment. In this case, the parameters µ 1

AP Statistics: 10.2: Comparing Two Means Name: Suppose we want to compare the average effectiveness of two treatments in a completely randomized experiment. In this case, the parameters µ 1 and µ 2 are

### CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

### 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

### Let s explore SAS Proc T-Test

Let s explore SAS Proc T-Test Ana Yankovsky Research Statistical Analyst Screening Programs, AHS Ana.Yankovsky@albertahealthservices.ca Goals of the presentation: 1. Look at the structure of Proc TTEST;

### Factorial Analysis of Variance

Chapter 560 Factorial Analysis of Variance Introduction A common task in research is to compare the average response across levels of one or more factor variables. Examples of factor variables are income

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

Statistics 101 106 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.

### Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

### Chapter 6: t test for dependent samples

Chapter 6: t test for dependent samples ****This chapter corresponds to chapter 11 of your book ( t(ea) for Two (Again) ). What it is: The t test for dependent samples is used to determine whether the

### 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

### Hypothesis Testing and Confidence Interval Estimation

Biostatistics for Health Care Researchers: A Short Course Hypothesis Testing and Confidence Interval Estimation Presented ed by: Susan M. Perkins, Ph.D. Division of Biostatistics Indiana University School

### 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

### 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

### AP STATISTICS 2009 SCORING GUIDELINES (Form B)

AP STATISTICS 2009 SCORING GUIDELINES (Form B) Question 5 Intent of Question The primary goals of this question were to assess students ability to (1) state the appropriate hypotheses, (2) identify and

### Inference for two Population Means

Inference for two Population Means Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison October 27 November 1, 2011 Two Population Means 1 / 65 Case Study Case Study Example

### Sampling and Hypothesis Testing

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

### 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

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

### 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.

### 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

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### 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

### Statistics 2014 Scoring Guidelines

AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### 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

### 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

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### 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

### 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

### Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to explain the difference between the p-value and a posterior

### 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

### Multiple Hypothesis Testing: The F-test

Multiple Hypothesis Testing: The F-test Matt Blackwell December 3, 2008 1 A bit of review When moving into the matrix version of linear regression, it is easy to lose sight of the big picture and get lost

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### 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

### Chapter 23 Inferences About 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

### Basic Statistics. Probability and Confidence Intervals

Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

### 6. Statistical Inference: Significance Tests

6. Statistical Inference: Significance Tests Goal: Use statistical methods to check hypotheses such as Women's participation rates in elections in France is higher than in Germany. (an effect) Ethnic divisions