Sociology 6Z03 Topic 15: Statistical Inference for Means


 Eugene Burke
 2 years ago
 Views:
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 singlepeaked, 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 tdistribution with n 1 degrees of freedom. Student was the nomdeplume of the discoverer of the tdistribution 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 tdistribution 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 tdistribution approaches the normal distribution. For small degrees of freedom, the tdistribution 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) tdistributions: 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 tdistribution 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 95percent 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 95percent 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 Pvalue for a ttest, but if we need to use the ttable we won t usually be able to find a precise Pvalue. Because the alternative hypothesis is directional, and specifies a positive value of µ, we find the Pvalue by looking in the righthand tail of the tdistribution with 9 degrees of freedom: OneSided 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 Pvalue for the hypothesis test? A.005 < P <.0025 B.0025 < P <.005 C P =.001 D I don t know. OneSided 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 Pvalue. t* = prob. to the right =.0025 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 t Hypothesis Test: Example, TwoSided Test Thought Question TRUE or FALSE: If the alternative hypothesis were nondirectional, H a : µ = 0, then, in comparison with the previous onesided test, we would double the Pvalue, which would be reported as t = 3.547, df = 9,.005 < P <.01, twotail. A TRUE B FALSE C I don t know. OneSided P TwoSided 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 ttests and tintervals is that in small samples they depend upon the assumption that the population is normally distributed. In large samples, ttests and tintervals are generally quite accurate even if the population is not normal, but in large samples inferences based on the tdistribution 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 stemplot for the illustrative dataset: There s nothing obviously problematic here: The distribution is singlepeaked, 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 tintervals and ttests for a single mean is for matchedpairs 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: BeforeAfter 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, matchedpairs designs tend to be more powerful than independentsamples 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 TwoSample ttests and tintervals 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 TwoSample ttests and tintervals Instead, we use the twosample tstatistic 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 tdistribution. John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41 Comparing Two Means From Independent Samples TwoSample t Confidence Interval To construct a levelc 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 tdistribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1 (or estimated by a complex formula, called the WelchSatterthwaite equation, given in the text). John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall / 41
17 Comparing Two Means From Independent Samples TwoSample 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 tdistribution with degrees of freedom equal to the smaller of n 1 1 and n 2 1 (or estimated by the WelchSatterthwaite 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 onesided or twosided 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 twosample 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 gradepoint averages of B or less; and (2) those with gradepoint 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: GradePoint 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 onesided 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 Pvalue for the test? A.025 < P <.05. B.01 < P <.02. C.005 < P <.01. D I don t know. OneSided P TwoSided 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 twosided alternative hypothesis, what would be the Pvalue 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. OneSided P TwoSided 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 nonnormal). 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: GradePoint 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
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay 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
More informationHypothesis 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
More informationStatistiek I. ttests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35
Statistiek I ttests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistieki/ John Nerbonne 1/35 ttests To test an average or pair of averages when σ is known, we
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationHypothesis 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 UrbanaChampaign Hypothesis Testing or How to Decide to Decide
More informationGeneral Method: Difference of Means. 3. Calculate df: either WelchSatterthwaite 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 WelchSatterthwaite formula or simpler df = min(n
More informationLAB 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.
More informationCONTENTS 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,
More informationConfidence 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
More informationModule 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
More informationStatistics 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
More informationStatistical 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
More informationSelected Nonparametric and Parametric Statistical Tests for TwoSample Cases 1
Selected Nonparametric and Parametric Statistical Tests for TwoSample Cases The Tstatistic is used to test differences in the means of two groups. The grouping variable is categorical and data for the
More information4. Introduction to Statistics
Statistics for Engineers 41 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
More informationLesson 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
More informationUCLA 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
More informationName: Date: Use the following to answer questions 34:
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
More informationGood 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
More informationComparing 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
More informationMind on Statistics. Chapter 13
Mind on Statistics Chapter 13 Sections 13.113.2 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
More informationStat 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 tdistribution as an approximation
More informationChapter 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
More informationTwosample ttests.  Independent samples  Pooled standard devation  The equal variance assumption
Twosample ttests.  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
More informationTwosample hypothesis testing, II 9.07 3/16/2004
Twosample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For twosample tests of the difference in mean, things get a little confusing, here,
More informationChapter 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
More informationNonparametric tests, Bootstrapping
Nonparametric tests, Bootstrapping http://www.isrec.isbsib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis
More informationInferences About Differences Between Means Edpsy 580
Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at UrbanaChampaign Inferences About Differences Between Means Slide
More informationConfidence 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
More information3.4 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
More informationAn Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10 TWOSAMPLE 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 TWOSAMPLE TESTS Practice
More informationSTAT 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
More informationNull Hypothesis Significance Testing Signifcance Level, Power, ttests Spring 2014 Jeremy Orloff and Jonathan Bloom
Null Hypothesis Significance Testing Signifcance Level, Power, ttests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is
More informationC. 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
More informationPermutation Tests for Comparing Two Populations
Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. JaeWan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of
More informationUnit 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
More informationAP 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
More informationPractice 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 12: 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,
More informationFairfield 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
More information3. 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
More informationSection 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
More informationThe Wilcoxon RankSum Test
1 The Wilcoxon RankSum Test The Wilcoxon ranksum test is a nonparametric alternative to the twosample ttest which is based solely on the order in which the observations from the two samples fall. We
More informationPart 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
More informationAP * 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
More informationHistogram. 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
More informationUnit 27: Comparing Two Means
Unit 27: Comparing Two Means Prerequisites Students should have experience with onesample tprocedures before they begin this unit. That material is covered in Unit 26, Small Sample Inference for One
More informationStudy 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
More informationMTH 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
More informationData 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
More informationIntroduction 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 midrange of how easy it is to use. Other options include SPSS,
More informationStatistics courses often teach the twosample ttest, linear regression, and analysis of variance
2 Making Connections: The TwoSample ttest, 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 twosample
More informationAP Statistics 2010 Scoring Guidelines
AP Statistics 2010 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in
More informationNCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, twosample ttests, the ztest, the
More informationOpgaven 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
More informationHypothesis Testing. Bluman Chapter 8
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 81 Steps in Traditional Method 82 z Test for a Mean 83 t Test for a Mean 84 z Test for a Proportion 85 2 Test for
More informationAnalysis 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:
More informationTwosample inference: Continuous data
Twosample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with twosample inference for continuous data As
More informationAn 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.
More information9.1 (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
More informationHow 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
More informationOutline. 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
More informationIntroduction 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
More informationSuppose 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
More informationCHAPTER 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
More informationHYPOTHESIS 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
More informationLet s explore SAS Proc TTest
Let s explore SAS Proc TTest Ana Yankovsky Research Statistical Analyst Screening Programs, AHS Ana.Yankovsky@albertahealthservices.ca Goals of the presentation: 1. Look at the structure of Proc TTEST;
More informationFactorial 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
More information1. 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. tdistributions.
More informationOutline. Definitions Descriptive vs. Inferential Statistics The ttest  Onesample ttest
The ttest Outline Definitions Descriptive vs. Inferential Statistics The ttest  Onesample ttest  Dependent (related) groups ttest  Independent (unrelated) groups ttest Comparing means Correlation
More informationChapter 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
More informationChapter 8 Introduction to Hypothesis Testing
Chapter 8 Student Lecture Notes 81 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
More informationHypothesis 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
More informationt Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon
ttests 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
More informationChapter 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
More informationAP 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
More informationInference 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
More informationSampling 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
More informationChapter 7. Oneway ANOVA
Chapter 7 Oneway ANOVA Oneway ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The ttest of Chapter 6 looks
More informationTutorial 5: Hypothesis Testing
Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrclmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................
More informationTesting Group Differences using Ttests, ANOVA, and Nonparametric Measures
Testing Group Differences using Ttests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 354870348 Phone:
More informationIntroduction 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.
More informationIntroduction. 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
More informationStatistical Inference and ttests
1 Statistical Inference and ttests Objectives Evaluate the difference between a sample mean and a target value using a onesample ttest. Evaluate the difference between a sample mean and a target value
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationStatistics 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
More informationConfidence 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
More informationLecture 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
More informationNCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, twosample ttests, the ztest, the
More informationObjectives. 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
More informationHypothesis 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
More informationComparison 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 pvalue and a posterior
More informationSTAT 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
More informationMultiple Hypothesis Testing: The Ftest
Multiple Hypothesis Testing: The Ftest 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
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationHYPOTHESIS 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
More informationChapter 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 300minute
More informationBasic 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
More information6. 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
More informationSample Exam #1 Elementary Statistics
Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables
More informationChapter 9, Part A Hypothesis Tests. Learning objectives
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
More information