...Business Statistics WISE-International Master Hypothesis Testing A hypothesis is a claim (conjecture/assumption) about a population parameter: population mean population proportion It is always about a population parameter, not a sample statistic A Common Theme Check the merits of this hypothesis based on sample information sample A hypothesis is formed about some population parameter infer Hypothesis testing provides a general framework for approaching such inference problems ˆ The Null Hypothesis Suppose that some hypothesis has been formed about the population parameter and that this hypothesis will be believed unless sufficient contrary evidence is produced. This hypothesis can be thought of as a maintained hypothesis. In the language of statistics, this hypothesis is called a null hypothesis, and is denoted as H0. In hypothesis testing, the null hypothesis plays a role similar to that of a defendant on trial in many judicial systems. Just as a defendant is presumed to be innocent until proven guilty, the null hypothesis is presumed to be true until the data strongly suggest otherwise. The Alternative Hypothesis, H1 Having a null hypothesis requires having an alternative hypothesis that challenges the null hypothesis. In a Court of Law H 0 : innocent H1 : guilty The defendant is deemed innocent until the prosecution presents sufficiently strong contrary evidence...
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...Hypothesis Testing Statistical Method Karl Phillip R. Alcarde MBA University of Negros Occidental-Recoletos DEFINITION DEFINITION Hypothesis testing or significance testing is a method for testing a claim or hypothesis about a parameter in a population, using data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a sample statistic could have been selected, if the hypothesis regarding the population parameter were true. Hypothesis testing or significance testing is a method for testing a claim or hypothesis about a parameter in a population, using data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a sample statistic could have been selected, if the hypothesis regarding the population parameter were true. The method of hypothesis testing can be summarized in four steps. 1. To begin, we identify a hypothesis or claim that we feel should be tested. For example, we might want to test the claim that the mean number of hours that children in the United States watch TV is 3 hours. 2. We select a criterion upon which we decide that the claim being tested is true or not. For example, the claim is that children watch 3 hours of TV per week. Most samples we select should have a mean close to or equal to 3 hours if the claim we are testing is true. So at what point do we decide that the discrepancy between the sample mean and 3 is so big that the claim we...
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...Chapter 11: Testing a Claim Objectives: Students will: Explain the logic of significance testing. List and explain the differences between a null hypothesis and an alternative hypothesis. Discuss the meaning of statistical significance. Use the Inference Toolbox to conduct a large sample test for a population mean. Compare two-sided significance tests and confidence intervals when doing inference. Differentiate between statistical and practical “significance.” Explain, and distinguish between, two types of errors in hypothesis testing. Define and discuss the power of a test. AP Outline Fit: IV. Statistical Inference: Estimating population parameters and testing hypotheses (30%–40%) B. Tests of significance 1. Logic of significance testing, null and alternative hypotheses; P-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power 4. Test for a mean (large sample -- ( known) What you will learn: A. Significance Tests for µ (( known) 1. State the null and alternative hypotheses in a testing situation when the parameter in question is a population mean µ. 2. Explain in nontechnical language the meaning of the P-value when you are given the numerical value of P for a test. 3. Calculate the one-sample z-statistic and the P-value for both one-sided and two-sided tests about the mean µ of a Normal population. 4. Assess statistical significance at standard levels α by comparing...
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...Applied Business Statistics FINAL Examination 1. The t distribution A) assumes the population is normally distributed. B) approaches the normal distribution as the sample size increases. C) has more area in the tails than does the normal distribution. D) All of the above. 2. Suppose a 95% confidence interval for μ turns out to be (1,000, 2,100). Give a definition of what it means to be "95% confident" in an inference. A) In repeated sampling, the population parameter would fall in the given interval 95% of the time. B) In repeated sampling, 95% of the intervals constructed would contain the population mean. C) 95% of the observations in the entire population fall in the given interval. D) 95% of the observations in the sample fall in the given interval. 3. In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a wider interval. A) narrower B) wider C) less significant D) biased 4. An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the upper end point in a 99% confidence interval for the average income? A) $15,052 B) $15,141 See worksheet C) $15,330 D) $15,364 5. A prison official wants to estimate the proportion of cases of recidivism....
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...take a sample. From the results of analysis from the sample data, we can predict the results from the population. Some questions that one may want to answer are 1. Are unmarried workers more likely to be absent from work than married workers? 2. Are the sixth graders in a certain school significantly less skilled in their mathematical abilities than the average student in the district? 3. In Fall 1996, did students in Math 163-01 score the same on the exam as students in Math 163-02? 4. Is there any difference between the strengths of steel wire produced by the XY Company and Bob’s Wire Company? 5. A hospital spokesperson claims that the average daily room charge for a specific procedure is $622. Can we reject this claim? WHAT IS A HYPOTHESIS? Hypothesis: A statement about the value of a population parameter developed for the purpose of testing. Examples of hypotheses, or statements, made about a population parameter are: The mean monthly income from all sources for systems analysts is $3,625. Twenty percent of all juvenile offenders ultimately are caught and sentenced to prison. Hypothesis testing: A procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected. lFollowing is a five-step procedure for testing a hypothesis. 5 STEPS IN THE HYPOTHESIS TESTING PROCEDURE 1. State the null hypothesis and the alternate hypothesis. Null Hypothesis – statement...
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...Department of Statistics Texas A&M University Using sample data to draw a conclusion about a population • Statistical inference provides methods for drawing conclusions about a population from sample data. • Two key methods of statistical inference: o o Confidence intervals Hypothesis tests (a.k.a., tests of significance) Hypothesis Testing: Evaluating the effectiveness of new machinery at the Bloggs Chemical Plant • Before the installation of new machinery, long historical records revealed that the daily yield of fertilizer produced by the Bloggs Chemical Plant had a mean μ = 880 tons and a standard deviation σ = 21 tons. Some new machinery is being evaluated with the aim of increasing the daily mean yield without changing the population standard deviation σ. Hypothesis Testing: Evaluating the effectiveness of new machinery at the Bloggs Chemical Plant Null hypotheses • The claim tested by a statistical test is called the null hypothesis. The test is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference”, that is, a statement of the status quo. Alternative hypotheses • The claim about the population that we are trying to find evidence for is the alternative hypothesis. The alternative hypothesis is one-sided if it states that a parameter is larger than or that it is smaller than the null hypothesis value. It is two-sided if it states that the parameter...
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...statistics calculated from samples to estimate the values of population parameters. Select Random Sample Sample for (statistic) Calculate to estimate Becomes Population Parameter. BASIC Example: Soft Drink Bottler μ=600, σ=10. Normal Distribution. What is P(X>598)? p(x<598) . Sampling Dist.of the Mean – Distribution of all Possible Sample Means if you select a sample of a certain size. μX= μ. μ = i=1NXiN (formula for mean) . σ = i=1N(Xi-μ)2N Although you do not know how close the sample mean of any particular sample selected comes to the pop mean, you know that the mean of all possible sample means that could have been selected = the pop mean. Standard error is calculating the probability of a certain amount of error. EXAMPLE: Standard error is . As n increases decreases. CENTRAL LIMIT THEOREM: Regardless of shape of individual values in distribution; as long as sample size is large enough the sampling distribution of the mean will be approximately normally distributed with μX= μ and σX= σ . For most population distributions n ≥ 30 will be large enough. For symmetric population distributions, n ≥ 5 is sufficient. For normal population distributions, the sampling distribution of the mean is always normally distributed EXAMPLE: SAMPLING DISTRIBUTION OF THE PROPORTION. π is the proportion of items in the population with a characteristic of interest. p is the sample proportion and provides an estimate of π. Underlying Sample distribution is binomial...
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...to take a sample. From the results of analysis from the sample data, we can predict the results from the population. Some questions that one may want to answer are 1. Are unmarried workers more likely to be absent from work than married workers? 2. Are the sixth graders in a certain school significantly less skilled in their mathematical abilities than the average student in the district? 3. In Fall 1996, did students in Math 163-01 score the same on the exam as students in Math 163-02? 4. Is there any difference between the strengths of steel wire produced by the XY Company and Bob’s Wire Company? 5. A hospital spokesperson claims that the average daily room charge for a specific procedure is $622. Can we reject this claim? WHAT IS A HYPOTHESIS? Hypothesis: A statement about the value of a population parameter developed for the purpose of testing. Examples of hypotheses, or statements, made about a population parameter are: The mean monthly income from all sources for systems analysts is $3,625. Twenty percent of all juvenile offenders ultimately are caught and sentenced to prison. Hypothesis testing: A procedure, based on sample evidence and probability theory, used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected. lFollowing is a five-step procedure for testing a hypothesis. 5 STEPS IN THE HYPOTHESIS TESTING PROCEDURE 1. State the null hypothesis and the alternate hypothesis. Null Hypothesis...
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...Elements of a Test of Hypothesis 1. Null Hypothesis (H0 ) - A statement about the values of population parameters which we accept until proven false. 2. Alternative or Research Hypothesis (Ha )- A statement that contradicts the null hypothesis. It represents researcher’s claim about the population parameters. This will be accepted only when data provides sufficient evidence to establish its truth. 3. Test Statistic - A sample statistic (often a formula) that is used to decide whether to reject H0 . 4. Rejection Region- It consists of all values of the test statistic for which H0 is rejected. This rejection region is selected in such a way that the probability of rejecting true H0 is equal to α (a small number usually 0.05). The value of α is referred to as the level of significance of the test. 5. Assumptions - Statements about the population(s) being sampled. 6. Calculation of the test statistic and conclusion- Reject H0 if the calculated value of the test statistic falls in the rejection region. Otherwise, do not reject H0 . 7. P-value or significance probability is defined as proportion of samples that would be unfavourable to H0 (assuming H0 is true) if the observed sample is considered unfavourable to H0 . If the p-value is smaller than α, then reject H0 . Remark: 1. If you fix α = 0.05 for your test, then you are allowed to reject true null hypothesis 5% of the time in repeated application of your test rule. 2. If the p-value of a test is 0.20 (say) and you reject H0 then, under...
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...Introduction to Hypothesis Testing 8.1 8.2 8.3 8.4 8.5 CHAPTER 8 Inferential Statistics and Hypothesis Testing Four Steps to Hypothesis Testing Hypothesis Testing and Sampling Distributions Making a Decision: Types of Error Testing a Research Hypothesis: Examples Using the z Test Research in Focus: Directional Versus Nondirectional Tests Measuring the Size of an Effect: Cohen’s d Effect Size, Power, and Sample Size Additional Factors That Increase Power LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 2 Identify the four steps of hypothesis testing. Define null hypothesis, alternative hypothesis, level of significance, test statistic, p value, and statistical significance. Define Type I error and Type II error, and identify the type of error that researchers control. Calculate the one-independent sample z test and interpret the results. Distinguish between a one-tailed and two-tailed test, and explain why a Type III error is possible only with one-tailed tests. Explain what effect size measures and compute a Cohen’s d for the one-independent sample z test. Define power and identify six factors that influence power. Summarize the results of a one-independent sample z test in American Psychological Association (APA) format. 8.6 3 4 5 8.7 8.8 8.9 8.10 SPSS in Focus: A Preview for Chapters 9 to 18 8.11 APA in Focus: Reporting the Test Statistic and Effect Size 6 7 8 2 PART III: PROBABILITY AND THE FOUNDATIONS OF INFERENTIAL STATISTICS 8.1 INFERENTIAL...
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...purpose of this paper is to review the hypothesis testing of the influence bank account balances have on ATM transactions. Team A will use a two-sample hypothesis test to compare two means to examine the hypothesis. The first part of the process will develop the hypothesis statement numerically and verbally. The test begins with the determination of the level of significance and decision rule whether to accept or not reject the null hypothesis. Calculating the test statistic will provide basis of the decision according to the critical region. The final section will describe the results in relation to the research question of bank account balance influencing the number of ATM transactions. Hypotheses Team A uses a five step hypothesis testing method as part of the research for this document in determining if two sample groups of bank account balances influence the number of ATM uses in one month. As with the steps for a one population hypothesis test, a two population hypothesis test follows the same sequence. Step one for Team A starts with stating the null hypothesis. Given the data shown in Appendix A and Appendix B, Team A has a hunch that bank account holders with larger balances tend to use ATMs more frequently than low account balance holders. Group A, from Appendix A, hold balances under $1600 for a given month and the Team measures the lower balance account’s ATM transactions during the given month. Group B samples include accounts with balances greater...
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...Hypothes9 9.1 Fundamentals of Hypothesis Testing: One-Sample Tests A Connection Between Confidence Interval Estimation and Hypothesis Testing Can You Ever Know the Population Standard Deviation? USING STATISTICS @ Oxford Cereals, Part II Fundamentals of Hypothesis-Testing Methodology The Null and Alternative Hypotheses The Critical Value of the Test Statistic Regions of Rejection and Nonrejection Risks in Decision Making Using Hypothesis Testing Hypothesis Testing Using the Critical Value Approach Hypothesis Testing Using the p-Value Approach 9.4 Z Test of Hypothesis for the Proportion The Critical Value Approach The p-Value Approach Potential HypothesisTesting Pitfalls and Ethical Issues 9.5 9.2 t Test of Hypothesis for the Mean (S Unknown) The Critical Value Approach The p-Value Approach Checking the Normality Assumption One-Tail Tests The Critical Value Approach The p-Value Approach 9.6 Online Topic: The Power of a Test USING STATISTICS @ Oxford Cereals, Part II Revisited CHAPTER 9 EXCEL GUIDE CHAPTER 9 MINITAB GUIDE 9.3 Learning Objectives In this chapter, you learn: • The basic principles of hypothesis testing • How to use hypothesis testing to test a mean or proportion • The assumptions of each hypothesis-testing procedure, how to evaluate them, and the consequences if they are seriously violated • How to avoid the pitfalls involved in hypothesis testing • Ethical issues involved in hypothesis testing U S I N G S TAT I S T I...
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...interval on either side of the observed statistic value is called the margin of error . Null hypothesis,- the claim assessed in a hypothesis test is called the null hypothesis usally the null hypotheses is a statement fo no change from the traditional value . Alternative Hypothesis proposes what we should conclude if we find the null hypothesis to be unlikely Statistically significance, when the p value falls below the alpha level we say that the test is statistically significance at the alpha level Alpha level, the threshold that determines when we reject a null hypothesis if we onserve a statistic whose p value based on the null hypothesis is less than a we reject that null hypothesis Significance level – the alpha level is also called the significancelevel most often in a phrase such as a conclusionthat a particular test is significanceat the 5% significance level . Type I error, the error of rejecting a null hypotheseswhen in fact it is true Type II Error, the errror of failing to reject a null hypothesis when in fact it is false P-Value population the entire group of individuals or instances about whom we hope to learn population parameter sample a subset of a population examined in the hope of learning about the population sample statistic simple random sample -a sampling design I which the population is divided into sevral subpopulation or strata and random samples are then drawn from each stratum . If the strata are homogeneous but are different from each...
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...GPA vs. Music OIS 2340 Section 1 Aubrey Bullough Alyssa Boyd Helena Paulos Moraya Dodson Sam Webster Executive Summary We are a radio station and we were interested in finding a new target group for our upcoming sister station and who better than to target the large student population? Our goal was to find out if listening to classical music while studying is beneficial to students’ GPA’s. We assumed that students who listened to classical music while studying have higher GPAs (3.1 or above) in comparison to students who listen to other genres while studying. A sample of 61 college students was taken and of those 16 reported that they listen to classical music while studying and obtain a GPA of 3.1 or above. We used proportional hypothesis testing to keep our data simple and precise. Our hypothesis was that 50% of students with GPAs 3.1 or above listened to classical music while studying. After extensive research and advanced calculations we have decided to reject this hypothesis. To gather our data we created a digital survey on Survey Monkey and had each member of our group post the survey link as their Facebook status. To our advantage, Facebook worked extremely quickly. Our questions in our survey consisted of: (a) “Do you listen to music while you study?” (b) “What genre do preferred to listen to while they study?” And (c) “What is your current GPA?” We received roughly 100 responses and of those 100, 61 students earned a GPA of 3.1 or above and listen to...
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... Contents CHAPTER 9: HYPOTHESIS TESTS 9.1 Elements of a Hypothesis Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Normal Hypothesis Test for Population Proportion p . . . . . . . . . . . . . . . . . . 9.3 The t-Test: Hypothesis Testing for Population Mean µ . . . . . . . . . . . . . . . . . 9.4 Possible Errors in Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Limitations and Common Misinterpretations of Hypothesis Testing . . . . . . . . . . 1 1 6 10 15 17 Stat 3011 Chapter 9 CHAPTER 9: HYPOTHESIS TESTS Motivating Example A diet pill company advertises that at least 75% of its customers lose 10 pounds or more within 2 weeks. You suspect the company of falsely advertising the benefits of taking their pills. Suppose you take a sample of 100 product users and find that only 5% have lost at least 10 pounds. Is this enough to prove your claim? What about if 72% had lost at least 10 pounds? Goal: 9.1 Elements of a Hypothesis Test 1. Assumptions 2. Hypotheses Each hypothesis test has two hypotheses about the population: Null Hypothesis (H0 ): Alternative Hypothesis (Ha ): 1 Stat 3011 Chapter 9 Diet Pill Example: Let p = true proportion of diet pill customers that lose at least 10 pounds. State the null and alternative hypotheses for the diet pill example. 3. Test Statistic Definition: Test Statistic A test statistic is a measure of how compatible the data is with the null hypothesis. The larger the test...
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