Comparing Two Means Hypothesis Testing

11 Key excerpts on "Comparing Two Means Hypothesis Testing"

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  eBook - PDF

  Statistics Unplugged

  • Sally Caldwell(Author)
  • 2012(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  From the formulation of a null hypothesis, to the identification of a critical value, through the conclusion and interpretation of results, you’ve traveled the road of hypothesis-testing basics. What’s more, you’ll likely discover that the research questions involving difference of means tests are very straightforward. The research situations that call for a difference of means test typically fall into two categories: situations involving matched or related samples, and situa-tions involving independent samples. That’s the order we’ll follow in this chapter. Before We Begin I’m going to make a suggestion that you may or may not appreciate. I’m going to ask you to go back to the previous chapter as soon as you finish this chapter. That’s right; first complete this chapter and then take another look at the previ-ous chapter. I know that may sound strange, but there’s a reason why I suggest it. I believe that many students of statistics have an easier time understanding hypothesis testing involving two samples than hypothesis testing involving one sample. As to why that might be the case, maybe it’s because the idea of compar-ing two groups (or more correctly, two samples) is such a commonplace activity, whether it is warranted or not. At any rate, my experience tells me that it is with the two sample applications that students typically start to get comfortable with hypothesis testing. Therefore, if you struggled with the last chapter or you still don’t feel on solid ground with it, let me urge you to move ahead with this chap-ter. Once you’re through, go back to the previous chapter. Related Samples To say that two samples are matched or related in some way is to say that the cases included in the samples were not selected independently of one another. Let me give you a few examples. Let’s say we give a group of students a drug awareness test, measuring their awareness of the dangers of recreational drug use.

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  Applied Medical Statistics

  • Jingmei Jiang(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  169 Applied Medical Statistics , First Edition. Jingmei Jiang. © 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc. Companion website: www.wiley.comgojiangappliedmedicalstatistics 8 Hypothesis Testing for Two Population Parameters CONTENTS 8.1 Testing the Difference Between Two Population Means: Paired Samples 170 8.2 Testing the Difference Between Two Population Means: Independent Samples 173 8.2.1 t-Test for Means with Equal Variances 173 8.2.2 F-Test for the Equality of Two Variances 176 8.2.3 Approximation t-Test for Means with Unequal Variances 178 8.2.4 Z-Test for Means with Large-Sample Sizes 181 8.2.5 Power for Comparing Two Means 182 8.2.6 Sample Size Determination 183 8.3 Testing the Difference Between Two Population Rates (Normal Approximation Method) 185 8.3.1 Power for Comparing Two Rates 186 8.3.2 Sample Size Determination 187 8.4 Summary 188 8.5 Exercises 189 In Chapter 7, we introduced the concepts of hypothesis testing and their application to one sample inference. The usage of one sample hypothesis testing, however, is limited in practice, as prespecified knowledge of the parameter is needed to establish the null and alternative hypotheses. Two sample inference is more common, regarding which underlying parameters of two populations are both unknown, and we are interested in whether the parameters are equal or not. Although the methods might differ based on the research design and the type of data, the principles and procedures are similar to those discussed in the previous chapter. We start with hypothesis testing for sample data from two populations of normal distribution or when the central limit theorem holds, and then, under this theorem, we discuss hypothesis testing for binominal data. Imagine that a researcher is planning a pilot experiment on the effect of a new anti-cancer drug using S180 tumor-bearing mice.

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  Understanding Political Science Statistics

  Observations and Expectations in Political Analysis

  • Peter Galderisi(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  HAPTER 7

  Hypothesis Testing and the Concept of Association

  Observations and Expectations about the Difference between Means 1

  CONTENTS

  • Comparison of Two Means
  • Special Comment on Significance Tests
  • Appendix: Comparison of Two Variables, Same or Matched Groups
  • Key Terms
  • Questions and Exercises

  Learning Objectives:

  • To continue our discussion of hypothesis testing
  • To understand how to apply the central limit theorem and normal curves to the analysis of differences between means
  • To understand and learn how to calculate the standard error of mean differences
  • To understand the difference between groups that are sampled independently and those in which the sampling in the second group is dependent on the first

  Thus far, we have acquainted ourselves only with descriptive and inferential measures of univariate statistics. The mathematical logic behind our next set of presentations on bivariate statistics follows the concepts already discussed. Before moving to an analysis of the association between two variables, however, some further terms need to be reintroduced.2

  We generally are concerned with analyzing bivariate (and eventually multivariate) statistics because we are trying to determine some causal linkage between two (or more variables). Let’s review some of the terms used in Chapter 1 . Perhaps we are trying to determine whether one’s gender affects one’s income in some expected way (with women earning less than men), or if race determines one’s chance of being selected for grand jury service (with minorities having a lower chance of selection than white Anglo-Americans), or if the type of voting system influences turnout (with proportional representation systems typically having higher turnout than single member districting systems). In the strictly bivariate case, we are trying to determine if differences in one variable (the causal agent or antecedent condition) affect the value of another (the outcome or consequence). The measured outcome, as you recall, is referred to as the dependent variable because its value is dependent on some causal or antecedent variable. The antecedent variable is called the independent variable. In our first example, the independent variable is gender (i.e., whether or not one is male or female); the dependent variable is income. In the second, the independent variable is minority (race or ethnicity) status, and the dependent variable is whether or not one is selected for jury service. In the third, the independent variable is the type of electoral system, and the dependent variable is turnout percentage. We would generally posit a guess about the relationship between these two variables. This guess, as we discussed in Chapter 1 , is called a hypothesis

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  Marketing Research

  • Carl McDaniel, Jr., Roger Gates(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Based on the result of the comparison, they either reject or fail to reject the null hypothesis. The t test also can be used to examine hypotheses about proportions from one sample or independent samples. When researchers need to test for differences among the means of three or more independent samples, analysis of vari- ance is the appropriate statistical tool. It is often used for hypoth- esis tests regarding the differences among the means of several independent groups. It permits the researcher to test the null hypothesis that there are no significant differences among the population group means. Key Terms analysis of variance (ANOVA) 305 Chi-square test 296 decision rule 290 degrees of freedom 295 F test 307 hypothesis 287 hypothesis test of proportions 302 independent samples 295 null hypothesis 288 p value 308 related samples 295 statistical significance 287 t test 299 type I error (α error) 290 type II error ( β error) 290 310 CHAPTER 13 Statistical Testing of Differences and Relationships Questions for Review & Critical Thinking 1. Explain mathematical differences, managerially important differences, and statistical significance. Can results be statisti- cally significant and yet lack managerial importance? Explain your answer. 2. Describe the steps in the procedure for testing hypotheses. Discuss the difference between a null hypothesis and an alterna- tive hypothesis. 3. Distinguish between a type I error and a type II error. What is the relationship between the two? 4. What is meant by the terms independent samples and related samples? Why is it important for a researcher to determine whether a sample is independent? 5. Your university library is concerned about student desires for library hours on Sunday morning (9:00 A.M.–12:00 P.M.). It has undertaken to survey a random sample of 1,600 undergraduate students (one-half men, one-half women) in each of four status levels (i.e., 400 freshmen, 400 sophomores, 400 juniors, and 400 seniors).

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  Statistical Concepts for the Behavioral Sciences

  • Harold O. Kiess, Bonnie A. Green(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  2. Explain why you cannot simply look at the difference between the mean scores of two groups given different treatments and decide if the difference between the means is due to the independent variable. 3. Calculate for each of the following sets of scores. s X 1 - X 2 Testing Your Knowledge 9.2 a. Group b. Group X 1 X 2 X 1 X 2 104 99 77 82 111 107 84 86 120 105 68 78 113 110 71 74 108 115 82 69 79 80 c. Group d. Group X 1 X 2 X 1 X 2 71 85 32 38 82 80 37 42 63 73 28 29 74 76 41 36 68 70 35 30 76 67 33 37 36 39 34 be close to zero, whereas if an independent variable has an effect and causes and to differ, then the value of t should become larger. Statistical Testing with t ind In using to test for a significant difference between two means, the steps for statistical hypothesis testing outlined in Chapter 8 are followed: 3 A null hypothesis, , and an alternative hypothesis, , are formulated. 3 The sampling distribution of t assuming is true is obtained. This distribution is given in Appendix C.2. 3 A significance level is selected. 3 A critical value of t, identified as , is found from the sampling distribution of t given in Appendix C.2. 3 Rejection regions are located in the theoretical sampling distribution of the t statistic. 3 The t statistic, identified as , is calculated from the sample data. 3 A decision to reject or not reject is made on the basis of whether falls into a rejection region. 3 The results of the statistical decisions are related to the research hypothesis. Statistical Hypotheses For a parametric test, a statistical hypothesis is a statement about population parameters. The parameters corresponding to a statistical test of and are the population means, and , respectively. For , the statistical hypotheses are (null hypothesis), (alternative hypothesis). For two populations of scores, one or the other of these hypotheses must be true.

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  Fundamental Statistics for the Behavioral Sciences

  • David Howell(Author)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In that example the same par-ticipants were observed both before and after the intervention. While that was a good way to evaluate the effects of this intervention pro-gram, in a great many experiments it is either impossible or undesirable to obtain data using repeated measurements of the same partici-pants. For example, if we want to determine if Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 352 Chapter 14 Hypothesis Tests Applied to Means: Two Independent Samples males are more socially inept than females, it clearly would be impossible to test the same people as males and then as females. Instead we would need a sample of males and a sec-ond, independent , sample of females. Probably the most common use of the t test is to test the difference between the means of two independent groups. We might want to compare the mean number of trials needed to reach criterion in a simple visual discrimination task for two groups of rats—one raised under normal conditions and one raised under conditions of sensory deprivation. Or in a memory study we might want to compare levels of retention for a group of college students asked to recall active declarative sentences and a group asked to recall passive negative sentences. As a final example, we might place participants in a situation in which another person needed help. We could compare the mean latency of helping behavior when participants were tested alone and when they were tested in groups.

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  New Statistical Procedures for the Social Sciences

  Modern Solutions To Basic Problems

  • Rand R. Wilcox(Author)
  • 2013(Publication Date)
  • Psychology Press

    (Publisher)

  CHAPTER 7 COMPARING TWO MEANS

  So far, hypothesis testing procedures have been limited to situations where a parameter is compared to a known constant. This problem occurs in practice, but a more common problem is comparing two or more unknown parameters to one another. This chapter considers the simplest of these problems, namely comparing the means of two normal distributions.

  7.1 COMPARING THE MEANS OF TWO INDEPENDENT NORMAL DISTRIBUTIONS

  It will help to keep a specific problem in mind, so suppose you want to compare two methods of teaching reading. A basic issue is whether it makes any difference, on the average, which teaching method is used. To find out, suppose you randomly assign n1 subjects to method 1 and n2 subjects to methods 2. After the course is completed, the subjects take a test that measures their reading ability. Let Xi j be the observed score of the ith subject in the jth group, i=1,…,nj ; j=1, 2.

  It is assumed that:

  1) X11 , X21 , …, , 1

  is a random sample from a normal distribution with mean μ1 and variance σ1 2 . These are the observations from group 1.

  2) X12 , X22 , …, , 2

  is a random sample from a normal distribution with mean μ2 and variance σ2 2 . These are the observations from group 2.

  3) The observations X11 , …, , 1

  are independent of X12 , …, , 2.

  4) .

  Comments about these assumptions are made at the end of this section.

  In the illustration, the goal is to determine whether there is any difference between the average reading scores for the subjects who receive teaching method 1 as opposed to method 2. Thus, the goal is to test

  Let and be the sample means corresponding to the two groups. By assumption, and are independent normal random variables, and so from Theorem 4.5.2,

  is normally distributed with mean

  and variance

  Let

  and let σ2 be the common value of σ1 2 and σ2 2 . That is, for notational convenience, let σ1 =σ2 =σ. Then

  and since E(s1 2 )=σ1 2 and E(s2 2 )=σ2 2 ,

  Thus, sy 2 is an unbiased estimate of σ2 . Also, (n1 -1)s1 2 /σ2 and (n2 -1)s2 2 /σ2 are independent chi-square random variables with n1 -1 and n2 -1 degrees of freedom. From section 6.1 , the sum of two independent chi-square random variables with ν1 and ν2 degrees of freedom is again a chi-square random variable, but with ν1 +ν2

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  Biostatistics

  A Foundation for Analysis in the Health Sciences

  • Wayne W. Daniel, Chad L. Cross(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  We wish to know if we have sufficient evidence to reject this hypothesis based on the data we have collected. Parametric Methods These methods are most often presented in introductory statistics textbooks, and hence tend to be reported quite often in the scientific literature. There are many assumptions associated with these 198 HYPOTHESIS TESTING tests, most of which have been explicated in previous chapters, and include issues of sample size, normality, equal variances, and so on. These assumptions exist because we are often testing pop- ulation parameters, namely, one or more mean (i.e., ) or proportion (i.e., p) using a test statistic that relies on these parameters and some measure of their error. All of these measurements require that data can be assumed to come from a defined statistical distribution (e.g., normal) that requires explicit assumptions be met. After calculating a statistic of interest, we wish to see how extreme our measurement is compared to that hypothesized under the null hypothesis. As in the illustra- tive example, we wish to know the magnitude of ( 1 −  2 ) and see if it is far enough away from 0 to reject the null hypothesis. The magnitude of our sample statistic provides information about the relative distant the actual mean difference is from the experiment given the null hypothesis. Considering a standard normal distribution, where 0 is in the center, large differences result in values far into the tails of the distribution. The area under the curve that exceeds this distance is called the p value. Hence, small p values result from large mean differences under a given null hypothesis, providing one measure of how incompatible our results are with our hypothesis. Parametric methods are often good choices under various circumstances, but sometimes data do not conform to the underlying assumptions of the tests of interest, and we must consider other procedures.

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  Mind on Statistics (with JMP Printed Access Card)

  • Jessica Utts, Robert Heckard(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Steps 4 and 5: Decide Whether the Result Is Statistically Significant Based on the p -Value and Report the Conclusion in the Context of the Situation These steps proceed as with any hypothesis test. Choose a significance level (“alpha”), usually .05, and declare the difference in means to be statistically significant if the p -value . Report a conclusion appropriate for the situation. In drawing a con-clusion about differences between two populations, it also is important to consider the way in which the data were gathered. Remember from Chapter 6 that data col-lected in an experiment can provide stronger evidence of a cause-and-effect relationship than data from an observational study. EXAMPLE 13.4 (cont.) The Effect of a Stare on Driving Behavior Step 1: State the hypotheses. The subscript 1 is used to represent the No Stare population, and the subscript 2 is used to represent the Stare population. As was stated above, the appropriate null and alternative hypotheses are H 0 : 1 2 0 (or 1 2 ) H a : 1 2 0 (or 1 2 ) Note that 1 = population mean time if everyone in the population of drivers were measured with no stare and 2 = population mean time if everyone in the population of drivers were measured while being stared at. The alternative hypothesis reflects the fact that the researchers thought staring would cause drivers to speed up, so the mean time should be slower (larger values) for the No Stare group and faster (smaller values) for the Stare group. SUPPLEMENTAL NOTE Equal Variances and Pooled Standard Error If we are willing to make the assump-tion that the two populations we are comparing have the same (or similar) variance, then there is a procedure for which the t -distribution is the correct distribution of the t -statistic when the null hypothesis is true. It involves pooling both samples to estimate the common variance. The procedure is called a pooled two-sample t -test , and it is discussed in Lesson 2 of this module.

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  Statistical Reasoning in the Behavioral Sciences

  • Bruce M. King, Patrick J. Rosopa, Edward W. Minium(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The result is s X− Y , the estimated standard error of the difference between sample means. To test differences between means, we assume homogene- ity of variance and obtain a pooled estimate of  X− Y with (n X − 1) + (n Y − 1) degrees of freedom. Protection from unfortunate consequences of violating the assumption of homogeneity of vari- ance is best assured by using large samples of equal size. We obtain a t value by dividing the deviation of ( X − Y ) from ( X −  Y ) hyp by the estimate of  X− Y : t = ( X − Y ) − ( X −  Y ) hyp s X− Y If the resultant t value falls in the region of rejection (which is determined by the nature of H A , the chosen value of , and the particular distribution of t related to df ), we reject H 0 and con- clude that the two samples were not drawn from two populations having the same mean. One of the basic assumptions that allow us to make inferences about population parameters from sample statistics is that of ran- dom sampling. In comparisons of two (or more) conditions, the random assignment of available subjects is often used as a substi- tute for random sampling. Although the t test can be used in this situation, statistics does not provide the basis for making gener- alizations beyond the exact subjects studied and the exact set of conditions at the time.

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  Introduction to Statistics and Data Analysis

  • Roxy Peck, Chris Olsen, , Tom Short, Roxy Peck, Chris Olsen, Tom Short(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  b. Suppose instead that each of the random samples had included 50 corn meal specimens. Explain why it would now be reasonable to carry out a two-sample t test. c. Assuming that each random sample size was 50, carry out a test to determine if there is a signifi-cant difference in mean fumonisin level for the two types of corn meal. Use a significance level of 0.01. 11.20 A researcher at the Medical College of Virginia con-ducted a study of 60 randomly selected male soccer players and concluded that frequently “heading” the ball in soccer lowers players’ IQs ( USA TODAY, August 14, 1995) . The soccer players were divided into two groups, based on whether they averaged 10 or more headers per game. Mean IQs were reported in the article, but the sample sizes and standard deviations were not Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 11.2 Inferences Concerning the Difference Between Two Population or Treatment Means Using Paired Samples 595 given. Suppose that these values were as given in the accompanying table. n Sample Mean Sample SD Fewer Than 10 Headers 35 112 10 10 or More Headers 25 103 8 a. Do these data support the researcher’s conclu-sion? Test the relevant hypotheses using a 5 0.05.

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