Mathematics
Hypothesis Testing
Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a hypothesis about the population parameter, collecting and analyzing data, and then determining whether the evidence supports or contradicts the initial hypothesis. The process typically involves setting up a null hypothesis and an alternative hypothesis, and using statistical tests to make a decision about the population parameter.
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9 Key excerpts on "Hypothesis Testing"
eBook - PDFProbability and Statistics
A Didactic Introduction
- José I. Barragués, Adolfo Morais, Jenaro Guisasola, José I. Barragués, Adolfo Morais, Jenaro Guisasola(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
CHAPTER 6 Tests of Hypotheses Martin Griffiths 1. Introduction to Hypothesis Testing One of the purposes of statistics is to enable us to make informed decisions based on some set of data. We will look here at a particular decision making process called Hypothesis Testing , an area of inferential statistics (in which we are trying to ‘infer’ something from the data). This aspect of statistics is often misunderstood, not only by students (Sotos, Vanhoof, Van den Noortgate and Onghena 2009) but also by scientists or other professionals who use statistical inference as part of their everyday work though are non-specialists in this area. The aim of this chapter is to introduce the ideas that lie behind Hypothesis Testing and to dispel the many misconceptions that arise commonly. We will keep this first section fairly non-technical for the basic ideas behind Hypothesis Testing to be absorbed. It is important to have the opportunity to see the ‘big picture’ before being concerned with the minutiae. Introductory Problem Suppose that a factory produces a particular electronic component that is used in mobile phones. In an efficiency drive, the engineer in charge of the production line is considering switching over to a new assembly procedure that he believes will reduce the time it takes to produce each component. He already knows, from measurements taken over several years, that when using the old procedure the mean production time for the component is 57 s and the standard deviation is 8 s. In order to test his hypothesis, he Mathematical Institute, University of Oxford (United Kingdom). Tests of Hypotheses 253 measured the time it took for each of 20 randomly-chosen components to be produced using the new procedure.
eBook - PDF- Ian C. Marschner(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
5 Hypothesis Testing concepts In Chapter 2 we discussed using confidence intervals to conclude whether or not particular values of the parameter are plausible. This introduced us to the notion of Hypothesis Testing, the principles of which will be developed in the next two chapters. In this chapter we will discuss some of the basic concepts in Hypothesis Testing, including significance level, power and P -values, and further discuss the connection between Hypothesis Testing and confidence intervals. Chapter 6 will then make use of these concepts to develop general methods for carrying out hypothesis tests based on the likelihood function. Our discussion in this chapter and the next will present hypothesis tests as rules for deciding whether an hypothesis should be rejected or not, based on the observed sample. This is useful for assessing the statistical significance of observed departures from the hypothesis, but reliance on a simple dichotomy can often lead to misinterpretation, so we will also discuss the importance of viewing Hypothesis Testing as complementary to the process of confidence interval estimation. These notions will be illustrated using an extended example involving randomised treatment comparisons. 5.1 Hypotheses In most biostatistical contexts involving Hypothesis Testing, one is interested in as-sessing whether there is evidence that a particular factor has an effect on some health outcome. In practice, the term “effect” is often used loosely to encompass both sit-uations that truly are the assessment of the effect of a factor, such as a randomised trial of an intervention, and situations involving only the assessment of associations, such as an observational study of a particular exposure. In either situation, statistical Hypothesis Testing begins with the development of a probability model, as was used in our discussions on parameter estimation.
- Ken Black, Tiffany Bayley, Ignacio Castillo(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
oberlo.com/blog/online-review-statistics. Introduction A key statistical mechanism for decision-making is the hypothesis test. The concept of Hypothesis Testing lies at the heart of inferential statistics, and the use of statistics to prove or disprove claims hinges on the concept. With Hypothesis Testing, business analysts are able to structure problems in such a way that they can use statistical evidence to test various theories about business phenomena. Business applications of statistical Hypothesis Testing run the gamut from determining whether a production line process is out of control to providing conclusive evidence that a new management leadership approach is significantly more effective than an old one. With the ever-increasing quantities and sources of data, the field of business analytics holds great potential for generating new hypotheses that will need to be tested and validated. Figure B.1 in Appendix B (Making Inferences About Population Parameters: A Brief Sum- mary) displays a tree diagram taxonomy of inferential techniques organized by usage, num- ber of samples, and type of statistic. While Chapter 8 contains the portion of these techniques that can be used for estimating a mean, a proportion, or a variance for a population with a single sample, Chapter 9 contains techniques used for testing hypotheses about a population mean, a population proportion, and a population variance using a single sample. The entire right side of the tree diagram taxonomy displays various hypothesis-testing techniques. The leftmost branch of this right side contains the Chapter 9 techniques (for single samples), and this branch is displayed in Figure 9.1. Note that at the end of each tree branch in Figure 9.1 the title of the statistical technique, along with its respective section number, is given for ease of identification and use.
eBook - PDF- D.G. Rees(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 10 Hypothesis Testing What tribunal can possibly decide truth in the clash of contradictory assertions and conjectures? 10.1 Introduction Statistical inference is concerned with how we draw conclusions from sample data about the larger population from which the sample has been selected. In the Chapter 9, we discussed one branch of inference, namely estimation, particularly confidence interval estimation. Another impor-tant branch of inference is Hypothesis Testing (Fig. 10.1), which is the subject of much of the remainder of this book. In this chapter we will consider again the five applications we looked at in Chapter 9 (see Sections 9-3, 9.4, 9*8, 9.10, and 9.11), but this time in terms of testing hypotheses about the various parameters. We end the chapter by discussing the connection between the two branches of infer-ence (see the dashed line in Fig. 10.1). The procedure for performing any hypothesis test can be set out in terms of seven steps: 1. Decide on a null hypothesis, H 0. 2. Decide on an alternative hypothesis, H 1. 3. Decide on a significance level. 4. Calculate the appropriate test statistic, using the sample data. 5. Find from tables the appropriate tabulated test statistic. 139 140 ■ Essential Statistics Statistical inference Confidence interval Hypothesis estimation testing Figure 10.1 Types of Statistical Inference 6. Compare the calculated and tabulated test statistics, and decide whether to reject the null hypothesis, F/0. 7. State a conclusion, after checking to see whether the assumptions required for the test in question are valid. Notes The steps above apply mainly to hypothesis tests performed ‘by hand’, for example, with a calculator and/or in an examination. If, on the other hand, we use Minitab for Windows to carry out a hypothesis test, steps 5 and 6 will be slightly different, as follows: Step 5.
eBook - PDFStatistical Methods
An Introduction to Basic Statistical Concepts and Analysis
- Cheryl Ann Willard(Author)
- 2020(Publication Date)
- Routledge(Publisher)
115 Hypothesis Testing Now that we have examined the properties of the normal distribution and you have an understanding of how probability works, let us look at how it is used in hypothesis test-ing. Hypothesis Testing is the procedure used in inferential statistics to estimate population parameters based on sample data. The procedure involves the use of statistical tests to deter-mine the likelihood of certain population outcomes. In this chapter, we will use the z -test, which requires that the population standard deviation ( σ ) be known. 8 Hypothesis Testing The material in this chapter provides the foundation for all other statistical tests that will be covered in this book. Thus, it would be a good idea to read through this chapter, work the problems, and then go over it again. This will give you a better grasp of the chapters to come. Tip! Hypothesis Testing usually begins with a research question such as the following: Sample Research Question Suppose it is known that scores on a standardized test of reading comprehension for fourth graders is normally distributed with μ = 70 and σ = 10. A researcher wants to know if a new reading technique has an effect on comprehension. A random sample of n = 25 fourth grad-ers are taught the technique and then tested for reading comprehension. A sample mean of M = 75 is obtained. Does the sample mean ( M ) differ enough from the population mean ( μ ) to conclude that the reading technique made a difference in level of comprehension? Our sample mean is, obviously, larger than the population mean. However, we know that some variation of sample statistics is to be expected just because of sampling error. What we want to know further is if our obtained sample mean is different enough from the popula-tion mean to conclude that this difference was due to the new reading technique and not just to random sampling error.
- Andrew F. Hayes(Author)
- 2020(Publication Date)
- Routledge(Publisher)
Type III errors are exceedingly rare, but they can’t be categorically ruled out. 8.3.2 The Validity and Power of a Statistical Test Hypothesis Testing is simply an exercise in probability computation and the use of the resulting p -value to make a decision about a research hypothesis translated into a statistical form. It is more accurate, however, to refer to Hypothesis Testing as an exercise in probability estimation . A distinction must be made between the true p -value and the estimated p -value . The true p -value is the actual probability of the obtained result or one more discrepant from the null hypothesis assuming the null hypothesis is true. The true p -value cannot be known exactly. The estimated p -value, in contrast, is the p -value that a Hypothesis Testing procedure produces. These are not necessarily the same thing. All the Hypothesis Testing procedures described in this book are theoretically derived attempts at estimating the true p -value. These tests typically require assumptions—conditions that must be met in order to produce a good estimate of the p -value. For example, the test described above for testing a hypothesis about a 179 8.4. Hypothesis Test or Confidence Interval? proportion assumes that the sample size is not small. If it is used with a small sample, the resulting estimated p -value can be inaccurate. (A similar test that does not make this assumption will be discussed in a later chapter.) The less accurate the test is at estimating the p -value, the more likely you are to make a decision error because the statistical decision is based on that p -value. Some statistical tests are better than others at accurately estimating the true p -value. Not all statistical tests are good, and even otherwise good Hypothesis Testing procedures may perform badly in some circumstances. 1 Statistical tests vary in what statisticians call a test’s validity .
- Richard J. Rossi(Author)
- 2022(Publication Date)
- Wiley(Publisher)
Note that the tail of a test is determined from the form of the alternative hypothesis, not the null hypothesis. In practice, the null and alternative hypotheses will be based on the goals of the research study. Example 7.3 One of the research phases in the development of a new drug is designed for investigating the side effects of the drug. In particular, the side effects of a new drug are usually compared with the side effects for a placebo treatment. A new drug is usually expected to have higher rates of side effects than the placebo, and therefore, it is common in a drug trial to test the ? 0 ∶ ? drug ≤ ? placebo versus ? A ∶ ? drug > ? placebo where ? drug and ? placebo are the proportions of individuals who will experience adverse side effects. Once a researcher has determined the null and alternative hypotheses that will be tested, a mechanism for weighing the evidence contained in the sample for or against the null hypothesis is needed. The statistic that is used to determine whether or not the observed sample provides evidence against a null hypothesis is called the test statistic . A test statistic is usually based on a point estimator of the parameter being tested and its sampling distri-bution is determined assuming that the null hypothesis is true. In particular, a test statistic compares the estimated value of a parameter with a hypothesized value of the parame-ter specified in the null hypothesis. For example, the test statistic that is used with large samples for testing ? 0 ∶ ? = 10 versus ? A ∶ ? ≠ 10 is ? = ̄ ? − 10 se ( ̄ ? ) The sampling distribution of a test statistic, assuming that ? 0 is true, is used to deter-mine the values of the test statistic for which ? 0 is to be rejected, and the set of values of the test statistic for which ? 0 is to be rejected is called the rejection region .
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
We begin with an unknown population, generally a population that has received a treatment. The question is to determine whether the treatment has an effect on the population mean (see Figure 8.2). 3. Hypothesis Testing is structured as a four-step process that is used throughout the remainder of the book. a. State the null hypothesis ( H 0 ), and select an alpha level. The null hypothesis states that there is no effect or no change. In this case, H 0 states that the mean for the treated population is the same as the mean before treatment. The alpha level, usually α = .05 or α = .01, provides a definition of the term very unlikely and determines the risk of a Type I error. Also state an alternative hypothesis ( H 1 ), which is the exact opposite of the null hypothesis. b. Locate the critical region. The critical region is defined as sample outcomes that would be very unlikely to occur if the null hypothesis is true. The alpha level defines “very unlikely.” For example, with α = .05, the critical region is defined as sample means in the extreme 5% of the distribu-tion of sample means. When the distribution is normal, the extreme 5% corresponds to z -scores beyond z = ± 1.96. c. Collect the data, and compute the test statistic. The sample mean is transformed into a z -score by the formula z 5 M 2 m s M The value of μ is obtained from the null hypoth-esis. The z -score test statistic identifies the location of the sample mean in the distribution of sample means. Expressed in words, the z -score formula is hypothesized sample mean − population mean z = standard error d. Make a decision. If the obtained z -score is in the critical region, reject H 0 because it is very unlikely that these data would be obtained if H 0 were true. In this case, conclude that the treatment has changed the population mean. If the z -score is not in the critical region, fail to reject H 0 because the data are not significantly different from the null hypothesis.
eBook - PDF- Ned Freed, Stacey Jones, Timothy Bergquist(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Chapter Exercises 341 GLOSSARY alternative hypothesis the position that will be embraced if the null hypothesis is rejected. critical value the cutoff point in a hypothesis test such that any sam- ple result beyond this cutoff will cause us to reject the null hypothesis. decision rule the rule used to either reject or fail to reject a null hypothesis given a specific sample result. null hypothesis the proposition to be tested directly in a hypothesis test. null sampling distribution the sampling distribution from which a sample result would be produced if the null hypothesis was true. one-tailed hypothesis test a test in which the concern is whether the actual value of the population parameter (for example, the popula- tion mean) is different—in a specific direction—from the value of the population parameter described in the null hypothesis. p-value measures the probability that, if the null hypothesis was true, we would randomly produce a sample result at least as unlikely as the one we’ve produced. significance level a specified probability that defines just what we mean by a sample result “so unlikely” under an assumption that the null hypothesis is true that such a result would cause us to reject the null hypothesis. significance test a hypothesis test. t test a hypothesis test using the t distribution. test statistic a descriptor of a given sample result to be used to decide whether to reject the null hypothesis. two-tailed hypothesis test a test in which the concern is whether the actual value of the population parameter (for example, the popula- tion mean) is different—in either direction—from the population parameter described in the null hypothesis. Type I error rejecting a true null hypothesis. Type II error accepting a false null hypothesis. G L OSS A R Y CHAPTER EXERCISES CHAPTER EXERCISES Critical value approach 54. Plexon Tire and Rubber Company claims that its Delton III tires have an average life of at least 50,000 miles.
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