Comparing Means in Two Populations

Size: px
Start display at page:

Transcription

1 Comparing Means in Two Populations

2 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 ll consider how to compare sample means from two populations. Towards the end of the course, we ll discuss comparing means from more than two populations. When we re comparing the means from two independent samples we usually ask: How does one sample mean compare with the other?

3 However, focusing just on comparing the means can be premature. It s safer to first consider the variability of each sample, the pattern of any outliers, the shape of the distributions. Then it may be safe to assume a normal distributions; but not always. So, we ll also discuss approaches to answering these questions when we re not comfortable with the assumption of normality, or when this assumption is just not defendable.

4 Two Sample Means Cities and counties: Returning to the Alabama SOL pass-rates, is there a difference between cities and counties? Recall last time we looked at a difference across years in the same population. Here, we want to look at the difference in one year between two populations: city high schools and county high schools.

5 Phase 1: State the Question, 1. Evaluate and describe the data Begin by looking at the data. Where did the data come from? What are the observed statistics? The source of this data is Alabama Department of Education. The first step in any data analysis is evaluating and describing the data. These first steps are also called preliminary analysis, to distinguish them from the definitive (or outcome) analysis.

6 Preliminary Analyses The goal of a preliminary analysis is to describe and inform. To give a description of the data. Keep in mind that, the goal of a definitive analysis is decision making or hypothesis testing.

7 Preliminary Analysis What are the observed statistics? Use the Fit Y by X platform to look at a graphical and tabular summary of the data, as in the next figure.

8

9 Note: Previously, in Step 1, we used the Distribution of Y reports to identify and fix errors and to further understand the data. You can use the Distn of Y but you would have to run it twice here you can see everything in one place. There are three components to this figure: The dot plots, the box plots, and the quantiles table. Let s look at each.

10 The Dot plot A dot plot shows the continuous Y-variable s (Algebra I 2000 pass rate) values along the vertical axis and the nominal X-variable s (City Yes or No) values along the horizontal axis. So we see the two groups along the horizontal axis; City = No and City = Yes. The width of the groups is proportional to the sample size of each group; there are more No (non-cities) values so it is drawn wider. This follows the your eye goes to ink rule. Groups with larger samples are more informative than groups with smaller samples so the larger n group is drawn bigger.

11 Dot plots Along the vertical axis, we see the 10th grade, year 2000 Algebra I SOL pass-rate; One dot for each high school. Values range from 0% passing to 100% passing. The horizontal spreading of the values is done so that you can see each school s scores better (called Jittered Points ). The amount of horizontal spread is random; so, don t try to interpret the scores for points farther to the right or left than scores closer to the center (horizontally). Of course, the vertical values are interpretable; that is, a school at the top has a higher pass-rate than a school at the bottom.

12 Box Plots These side-by-side box plots describe the shape of the distributions within each group. These plots do not assume normality so we use them to begin to answer the question, Is each group normally distributed? In the box plots we can easily see whether the values are symmetric about the median. Look for these warning flags that the data is not normal:

13 Box Plots Is the distance between the median and the 75%tile different than the distance between the median and the 25%tile? Is the upper whisker-bar (actually, the 90%tile) more distant from the median than the lower whisker-bar (the 10%tile)? Are the high-extreme tail-values more distant from the median than the low-extreme tail-values? These informal, graphic assessments don t raise any warning flags for these data. The dotted horizontal line represents the mean value for all schools (not considering the group).

14 Quantiles Report If a more detailed comparison of values is needed, the numerical values plotted in the box plots are shown in the Quantiles Report.

15 Quantiles Report For instance, in the City = No group, the distance from the median to the 75%tile (~49 vs 65, 16 points) is about the same as the distance between the median and the 25%tile (~49 vs 34, about 15 points). But, as in the Distribution platform, the preferred way to answer the question, Is each group normally distributed? is with a normal quantile plot.

16 Normal Quantile Plots Actually, the more proper phrasing of the question is: Within each group, is each group normally distributed? That is, it may be that if we were to lump both groups together, the data would appear non-normal. We must take group membership into account when making this assessment.

17

18 Interpretation Follow the same interpretation of the normal quantile plot as we discussed with a single mean. Are each group of black dots along a straight line? In the SOL data, these two sets of points follow the lines fairly well, with some departure in the tails.

19 Normality So, we now have enough information to answer the question, Within each group, is each group normally distributed? If the answer is Yes or Probably then we can proceed with parametric tests to compare the means. The Central Limit Theorem can apply if the sample size is large. The rule of thumb is if the total n is at least 30 (n1 + n2 30). If the answer to the normality question is No or I doubt it then we ll use nonparametric methods to answer our question.

20 Preliminary analysis, showing means If the data is normally distributed then means and SDs make sense. If these distributional assumptions are unwarranted, then we should consider nonparametric methods. Thus, the next thing to do in our preliminary analysis may be to get rid of the box plots and quantile plot and to show the means and standard deviations calculated within each group.

21

22 Means This figure shows the means, here connected with a line, and a short dashed bar that is one standard-error error bars. The long dashed lines above and below the means are one standard deviation away from their respective mean. The means and standard errors and deviations can be shown by selecting Means and Std Dev from the main Options menu. From the Display Options sub-menu in the Options menu, select the options necessary.

23 Means and SDs Report We can use it to describe the following: the number of observations in each group, the means of each group, the standard deviation within each group. Recall that the SE is not a descriptive statistic for the data, it is used for inference about the mean. JMP includes the SE here because it is used to form confidence intervals about the mean.

24 Note: You can change the number of decimal places displayed in any JMP report: Double-click a number in the report. A dialog will appear. Change the number of decimal places.

25 Summary: Preliminary Analysis So far, what have we learned about the data? We have not found any errors in the data. We re comfortable with the assumption of normality within each group. We ve obtained descriptive statistics for each of the group we re comparing.

26 Preliminary Results Also, at this point, we can look at the two means and make a guess, is there a difference between cities and counties? City schools seem to be about 12 points below non-city schools, and with SEs < 3, this seems like a big difference. Recall that the t-statistic is the ratio of the difference to a standard error. The ratio of 12 to 3 is bigger than 2.

27 2. Review assumptions As always there are three questions to consider: Is the process used in this study likely to yield data that is representative of each of the two populations? Yes, it is the population Is observation in the two samples independent of the others? Yes. Is the sample size sufficient? Yes, both groups are large and we re comfortable with normality for both groups.

28 Bottom line We have to be comfortable that the first two assumptions are met before we can proceed at all. If we re comfortable with the normality assumption, then we proceed, as below. Later, we ll discuss what to do when normality can not be safely assumed.

29 3. State the question in the form of hypotheses Let s refer to the two groups as 1 and 2 for notational purposes. Using these as subscripts, there are three possible null hypotheses: 1. The null hypothesis is µ1 µ2, 2. The null hypothesis is µ1 µ2, or 3. The null hypothesis is a fixed value, µ1 = µ2. And the alternative hypothesis is the opposite of the null.

30 test statistic = summary statistic - hypothesized paramter standard error of the summary statistic Phase 2: Decide How to Answer the Question 4. Decide on a summary statistic that reflects the question Recall the general test statistic: test statistic = summary statistic - hypothesized paramter standard error of the summary statistic

31 Difference Score In this situation (as in comparing paired means), we are going to use the difference score as our summary statistic: The relevant statistic is y1 y2 or the observed difference of the two means.

32 The hypothesized parameter is easy: µ 1 µ 2 = 0, since under any of the three null hypotheses a difference of 0 would result in failing to reject the null hypothesis.

33 Standard Error What about the standard error? There are two possibilities for the standard error y y of 1 2. The two possibilities depend upon the two standard deviations within each group. o Are they the same? o Or do the two groups have different standard deviations?

34 Same SD If the standard deviations (or variances) within the two populations are equal than the standard error of the difference is easy. We just average the two estimated standard deviations and obtained a pooled estimate. The variance of the mean difference is the sum of the standard errors of each mean

35 Same SD σ 2 σ n n 1 2

36 The t-statistic We ll use the t-test to compare the two sample means and, using a pooled estimate for the variance called 2 s p, we calculate: t = y s y p sp n + n 1 2

37 Estimating σ The pooled variance estimate is a weighted average of the two individual-group variances: s 2 p = ( 1) + ( 1) n s n s n + n Under the equal variance assumption, we calculate the p-value using df = n1 + n2 2.

38 Unequal SD If the variances are not equal, the calculation is more complicated: t = y s y s2 n + n 1 2

39 t-prime Note that the separate variance estimates are used in this t prime statistic, not the pooled estimated for variance. Further, the df is not a simple function of just n1 and n2. The details of these calculations are not important. What we need to know is how to proceed using JMP.

40 Deciding on the correct t-test Which test should we use? We may not need to choose; if the two sample sizes are equal (n1 = n2) the two methods give identical results. It s even pretty close if the n s are slightly different. If one n is more than 1.5 times the other (in the SOL case, n1 = 306 and n2 = 93, which is over 3 times as large), you ll have to decide which t-test to use.

41 Decision Decide whether the standard deviations are different. Use the equal variance t-test if they are the same, or Use the unequal variance t-test if they are different. Or, you could decide not to decide; use the unequal variance t-test. It s more conservative.

42 Determining Equal SDs There are three ways to make this decision. 1. Inspect the two standard deviation estimates. 2. Use the normal quantile plot. 3. Test for equal standard deviations.

43 Inspect the SDs Refer to the Means and Std Deviations report. Look at the two standard deviations, in this case 22.1 and Form the ratio of the largest to the smallest. If the ratio is larger than about 3, then the two SD s may be unequal (in our case, the ratio is 3.3). For a better answer to the question, see the normal quantile plot.

44 Normal Quantile Plot, SDs If the two standard deviations are equal then the slopes for the two lines in the normal quantile plot will be the same (the lines will be parallel). In our case the lines have roughly the same slope. So, for the SOL data the assumption of equal variability seems safe.

45 Questionable Parallel? If the slopes of the two lines are in that gray area between clearly parallel and clearly not parallel, what do we do? There are four possibilities: 1. Ignore the problem and be risky: use the equal variance t-test. 2. Ignore the problem and be conservative: use the unequal variance t-test. 3. Make a formal test of unequal variability in the two groups. 4. Compare the means using nonparametric methods.

46 What if not Parallel? Here, the data appear to be reasonably normal (this isn t the question) but the lines are not parallel they start out close and end far apart.

47 Here, not only do the variances appear to be unequal but normality is also in questions. We ll look at this case later.

48 Test for equal SDs In the case of the first figure, where we ll be using a t-test, but we re not sure which one, JMP provides a way to test for equal variance in the main options menu.

49

50 Choosing between the tests of equal variance Of the five tests: O Brien s, the Brown-Forsythe test, Levene s test, Bartlett s test, and the F-test; the last three are out of date and are not recommended. There s not much difference between O Brien s and Brown-Forsythe. Brown-Forsythe is more robust (resistant to outlying observations), so we ll use this result. What are we testing?

51 Variance test The null-hypothesis for these tests are, the variances are equal. So, if the Prob>F value for the Brown- Forsythe test is < 0.05, then you will reject the null hypothesis (universal decision rule) and conclude that the groups have unequal variances.

52 Reject Equal SDs? The report also shows the result for the t-test to compare the two means, allowing the standard deviations to be unequal. This is the unequal variance t-test. Here is a written summary of the results using this method: The two groups were compared using an unequal variance t-test and found to be significantly different (t = 7.1, df = 217.4, p-value < ). School districts in cities had lower scores.

53 Unequal test df? Notice the degrees of freedom it isn t a whole number. That is because it is based on a weighted contribution of each sample (with unequal ns) to the standard error estimate. You can round the number off but ONLY to one decimal place, do not round to a whole number.

54 Step 5: Random Variation Recall the rough interpretation that t s larger than 2 are likely not due to chance. For either type of t

55 6. State a decision rule The universal decision rule Reject H0: if p-value < a.

56 Phase 3: Answer the Question 7. Calculate the statistic There are three possible statistics that may be appropriate: 1. an equal variance t-test, 2. an unequal variance t-test, or 3. the nonparametric Wilcoxon rank-sum test.

57 Equal variance If the equal variance assumption is reasonable, then the standard t-test is appropriate. Note: When reporting a t-test it s assumed that, unless you specify otherwise, it s the equal-variance t-test. The next figure shows the means diamonds in the dot plot, the t-test report, and the means for a oneway ANOVA (Analysis Of VAriance) report. We ll cover oneway ANOVA later in the course. When there are only two groups, the t-test and ANOVA give identical results.

58

59 In JMP To compare the two means using an equal variance t- test in JMP: Choose Means/Anova/Pooled t from the main options menu. This adds the Oneway ANOVA report and means diamonds. The t-value, df, and p-value are shown in the t-test report. However, only the two-tailed p-value is reported (under Prob> t ). If the null-hypothesis specified that we were testing for equality, then this is the p-value we want.

60 One-tail p-values First, which group did JMP use for y1 and which for y2? JMP uses the order of the X-variable: If the X-variable is character, JMP alphabetically sorts the values and whichever comes first is y1. If the X-variable is numeric, JMP uses the smallest value of the X-variable as y1.

61 If your alternative was H A : µ 1 > µ 2 and the t-test value is positive, then the onetailed p-value is half the two-tail Prob> t in the report. If the t-test value was negative then you ve observed a difference in the opposite direction from that expected. The p-value is one minus half the Prob> t in the report.

62 If your alternative was H A : µ 1 < µ 2 and the t-test value is negative, then the onetailed p-value is half the two-tail Prob> t in the report. If the t-test value was positive then you ve observed a difference in the opposite direction from that expected. The p-value is one minus half the Prob> t in the report.

63 Unequal variance If the variances are not equal or if we just want a more conservative test then see the bottom portion of the Tests that the Variances are Equal report. The unequal-variances t-test is listed as the Welch Anova. Report the t-value, df and p-value, as in the equal variance case.

64 Nonparametric comparison of the medians If normality isn t reasonable, then you can use a nonparametric test. You will use a test that compares the medians between the two groups. The nonparametric test is based solely on the ranks of the values of the Y-variable. In JMP, choose Nonparametric > Wilcoxon test.

65 Wilcoxon The Wilcoxon rank-sum test (also called the Mann-Whitney test) ranks all the Y-values (in both groups) and then compares the sum of the ranks in each group (the groups are specified by the X-variable). If the median of the first group is, in fact equal to the median of the second group, then the sum of the ranks should be equal for equal sample sizes

66

67 Reporting Wilcoxon When reporting the results of a nonparametric test, it s usual to only report the p-value. In the above report, there are two p-values, one for the z-test and one using a chi-square value. The p-values will rarely be different. For large samples report the p-value from the normal approximation. For smaller samples, use the chi-square For really small samples you should probably consult a statistician to help you obtain exact p-values.

68 Steps 8 & 9 8. Make a statistical decision Using all three tests, the two groups are different. All p-values are < State the substantive conclusion The schools in cites have significantly lower mean pass rates (35.9% vs 48.8%) and significantly different median pass rates (36.0% vs 49.2%)

69 Phase 4: Communicate the Answer to the Question 10. Document our understanding with text, tables, or figures The year 2000 Alabama SOL pass-rates in 10th grade Algebra I were divided into two groups according to whether the school was a city or county high school. There were n = 306 schools within city school-districts and n = 92 in county school districts. The observed average pass rates within city schools was 35.9% (SD = 19.9) and pass rates outside of cities were 48.8% (SD = 22.1). Using a two-tailed t-test, we conclude that the observed means are significantly different (t = 5.0, df = 396, p-value < ). From this we conclude that city schools have a significantly lower pass rate compared to county schools. The 95% confidence interval about the mean difference is between 7.8% and 17.9%.

70 Less text, replaced by information in a table Alternatively, it may be more straightforward to include many of the numbers in a table and state your results in text. So instead of the above paragraph, you could do this:

71 The summary results for the year 2000 Alabama SOL pass rate percentages in 10th grade algebra I are shown in Table 1. Schools were divided into cities if their district name contained City and were otherwise classified as a county school district. From these results we conclude that schools in cities had a pass-rate that was significantly lower than the pass-rates compared to county schools. The 95% confidence interval about the mean difference is between 7.8% and 17.9%.

72 Table Alabama SOL Pass-Rates in 10 th Grade Algebra I For Schools in Cities and in Counties Location Number of Schools Pass Rate (SD) SE 95% CI City (19.90) County (22.05) Difference 12.9 * * t = 5.0, df = 396, p-value <

73 Less text, replaced by information in a figure As another alternative, it may be more informative to describe the results in a figure. Instead of the above, you could do this:

74 The summary results for the year 2000 Alabama SOL pass rate percentages in 10th grade algebra I are shown in Figure 12. Schools were divided into city and county high schools. From these results we conclude that schools in cities had a pass-rate that was significantly lower than the pass-rates compared to county schools(t = 5.0, df = 396, p-value < ). The 95% confidence interval about the mean difference is between 7.8% and 17.9%.

75 County High Schools City High Schools mean = 48.8, SD = mean = 35.9, SD = Figure Alabama SOL Pass-Rates in 10 th Grade Algebra I For Schools in Cities and in Counties

76 Summary: Two Independent Means Briefly, here is how to proceed when comparing the means obtained from two independent samples. Describe the two groups and the values in each group. What summary statistics are appropriate? Are there missing values? (why?) Assess the normality assumption. If normality is not warranted, then do a nonparametric test to compare the medians. If normality is warranted, then assess the equal variance assumption.

77 Summary (cont) Report confidence intervals on each of the means if normality is reasonable. Perform the appropriate statistical test: equal variance t-test, unequal variance t-test, or the Wilcoxon rank-sum test. Determine the p-value that corresponds to your hypothesis. Reject or fail to reject? State your substantive conclusion.

78 Summary (cont) Additional note: Say you conclude that the groups have different means. How do you describe what the different means are? If you ve followed the above recipe, you are in one of three situations: 1. Normality is reasonable and the variances are equal 2. Normality is reasonable and the variances are unequal 3. Normality is not warranted

79 Summary (cont) Normality is reasonable and the variances are equal, use the equal variance t-test. The write up reads the means are significantly different (t = x.xx, df =xxx, p-value = 0.xxxx). Also give a table of means, SEs and 95%CIs just like the Means for Oneway Anova.

80 Summary (cont) The variances are unequal, use unequalvariance t-test. The write up reads the means are significantly different (unequal variance t = x.xx, df =xxx, p-value = 0.xxxx). Also give a table of means, SEs and 95%CIs just like the Means and Std. Deviations report. Note: the means are the same. The SEs and CIs are different.

81 Summary (cont) Normality is unreasonable, use Wilcoxon s test. The write up reads the medians are significantly different (by Wilcoxon s signed-rank test, p-value = 0.xxxx). Also give a table of medians and IQRs. There is a way to put 95%CIs on these estimates but not using any easily available software.

82 Always Report a measure of the center and spread. For all three tests, report the p-value and make a decision based upon your hypothesis. Your final statement should summarize the results in terms of the experiment (no statistics),

Analysis of Variance ANOVA

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

SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

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

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

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

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

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

Simple Regression Theory II 2010 Samuel L. Baker

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

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

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

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

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.

THE KRUSKAL WALLLIS TEST

THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2 NON

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

1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

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,

Non-Parametric Tests (I)

Lecture 5: Non-Parametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of Distribution-Free Tests (ii) Median Test for Two Independent

NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

NCSS Statistical Software. One-Sample T-Test

Chapter 205 Introduction This procedure provides several reports for making inference about a population mean based on a single sample. These reports include confidence intervals of the mean or median,

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

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

DATA INTERPRETATION AND STATISTICS

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

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

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

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

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

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

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

Descriptive Statistics

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

SPSS Explore procedure

SPSS Explore procedure One useful function in SPSS is the Explore procedure, which will produce histograms, boxplots, stem-and-leaf plots and extensive descriptive statistics. To run the Explore procedure,

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

6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

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

The Dummy s Guide to Data Analysis Using SPSS

The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests

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:

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

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

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

13: Additional ANOVA Topics. Post hoc Comparisons

13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Kruskal-Wallis Test Post hoc Comparisons In the prior

How To Check For Differences In The One Way Anova

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

Once saved, if the file was zipped you will need to unzip it. For the files that I will be posting you need to change the preferences.

1 Commands in JMP and Statcrunch Below are a set of commands in JMP and Statcrunch which facilitate a basic statistical analysis. The first part concerns commands in JMP, the second part is for analysis

Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck!

Psychology 60 Fall 2013 Practice Exam Actual Exam: Next Monday. Good luck! Name: 1. The basic idea behind hypothesis testing: A. is important only if you want to compare two populations. B. depends on

2.2 Derivative as a Function

2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test

Nonparametric Two-Sample Tests Sign test Mann-Whitney U-test (a.k.a. Wilcoxon two-sample test) Kolmogorov-Smirnov Test Wilcoxon Signed-Rank Test Tukey-Duckworth Test 1 Nonparametric Tests Recall, nonparametric

HYPOTHESIS TESTING WITH SPSS:

HYPOTHESIS TESTING WITH SPSS: A NON-STATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER

StatCrunch and Nonparametric Statistics

StatCrunch and Nonparametric Statistics You can use StatCrunch to calculate the values of nonparametric statistics. It may not be obvious how to enter the data in StatCrunch for various data sets that

Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

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

Testing for differences I exercises with SPSS

Testing for differences I exercises with SPSS Introduction The exercises presented here are all about the t-test and its non-parametric equivalents in their various forms. In SPSS, all these tests can

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

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

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

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

CALCULATIONS & STATISTICS

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

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

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

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

Rank-Based Non-Parametric Tests

Rank-Based Non-Parametric Tests Reminder: Student Instructional Rating Surveys You have until May 8 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs

Paired T-Test. Chapter 208. Introduction. Technical Details. Research Questions

Chapter 208 Introduction This procedure provides several reports for making inference about the difference between two population means based on a paired sample. These reports include confidence intervals

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

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

Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk

Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk Structure As a starting point it is useful to consider a basic questionnaire as containing three main sections:

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

Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

Statistics for Sports Medicine

Statistics for Sports Medicine Suzanne Hecht, MD University of Minnesota (suzanne.hecht@gmail.com) Fellow s Research Conference July 2012: Philadelphia GOALS Try not to bore you to death!! Try to teach

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

Chapter 2. Hypothesis testing in one population

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

Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

Permutation & Non-Parametric Tests

Permutation & Non-Parametric Tests Statistical tests Gather data to assess some hypothesis (e.g., does this treatment have an effect on this outcome?) Form a test statistic for which large values indicate

Statistics. One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples

Statistics One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples February 3, 00 Jobayer Hossain, Ph.D. & Tim Bunnell, Ph.D. Nemours

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

Simple Linear Regression Inference

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

Skewed Data and Non-parametric Methods

0 2 4 6 8 10 12 14 Skewed Data and Non-parametric Methods Comparing two groups: t-test assumes data are: 1. Normally distributed, and 2. both samples have the same SD (i.e. one sample is simply shifted

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....................

Non-Inferiority Tests for Two Means using Differences

Chapter 450 on-inferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for non-inferiority tests in two-sample designs in which the outcome is a continuous

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

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

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

1 Nonparametric Statistics

1 Nonparametric Statistics When finding confidence intervals or conducting tests so far, we always described the population with a model, which includes a set of parameters. Then we could make decisions

Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion

Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

UNDERSTANDING THE TWO-WAY ANOVA

UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

2 Sample t-test (unequal sample sizes and unequal variances)

Variations of the t-test: Sample tail Sample t-test (unequal sample sizes and unequal variances) Like the last example, below we have ceramic sherd thickness measurements (in cm) of two samples representing

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

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

Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

SPSS Tests for Versions 9 to 13

SPSS Tests for Versions 9 to 13 Chapter 2 Descriptive Statistic (including median) Choose Analyze Descriptive statistics Frequencies... Click on variable(s) then press to move to into Variable(s): list

Statistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl

Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic

Difference tests (2): nonparametric

NST 1B Experimental Psychology Statistics practical 3 Difference tests (): nonparametric Rudolf Cardinal & Mike Aitken 10 / 11 February 005; Department of Experimental Psychology University of Cambridge

Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

Using Excel for inferential statistics

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

How To Test For Significance On A Data Set

Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

Friedman's Two-way Analysis of Variance by Ranks -- Analysis of k-within-group Data with a Quantitative Response Variable

Friedman's Two-way Analysis of Variance by Ranks -- Analysis of k-within-group Data with a Quantitative Response Variable Application: This statistic has two applications that can appear very different,

Projects Involving Statistics (& SPSS)

Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,

Hypothesis testing - Steps

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

UNDERSTANDING THE INDEPENDENT-SAMPLES t TEST

UNDERSTANDING The independent-samples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly

individualdifferences

1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,

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

Come scegliere un test statistico

Come scegliere un test statistico Estratto dal Capitolo 37 of Intuitive Biostatistics (ISBN 0-19-508607-4) by Harvey Motulsky. Copyright 1995 by Oxfd University Press Inc. (disponibile in Iinternet) Table

Experimental Designs (revisited)

Introduction to ANOVA Copyright 2000, 2011, J. Toby Mordkoff Probably, the best way to start thinking about ANOVA is in terms of factors with levels. (I say this because this is how they are described

Stat 5102 Notes: Nonparametric Tests and. confidence interval

Stat 510 Notes: Nonparametric Tests and Confidence Intervals Charles J. Geyer April 13, 003 This handout gives a brief introduction to nonparametrics, which is what you do when you don t believe the assumptions

Introduction. Statistics Toolbox

Introduction A hypothesis test is a procedure for determining if an assertion about a characteristic of a population is reasonable. For example, suppose that someone says that the average price of a gallon

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

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

T O P I C 1 2 Techniques and tools for data analysis Preview Introduction In chapter 3 of Statistics In A Day different combinations of numbers and types of variables are presented. We go through these

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

Hypothesis Testing for Beginners

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