Technology & Engineering
Hypothesis Test for a Population Mean
A hypothesis test for a population mean is a statistical method used to make inferences about the true mean of a population based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting sample data, calculating a test statistic, and determining the likelihood of observing the sample mean if the null hypothesis is true. This test helps to assess whether there is enough evidence to support a claim about the population mean.
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eBook - ePub- David M. McEvoy(Author)
- 2018(Publication Date)
- Wiley(Publisher)
Chapter 8 Hypothesis Tests of a Population MeanIn 2010, a business professor at the University of Central Florida confronted the students in his strategic management course about cheating on a mid-term exam.1 The class was huge, with over 600 students. The professor was convinced that roughly 200 students had cheated by getting the answers to the exam in advance. The interesting part is that he had no direct evidence. The exams were proctored in a laboratory environment and not a single student was actively caught cheating. Rather, the professor relied on statistics to drive his initial suspicion. The professor had been teaching the class for many semesters and as part of the discovery process, he conducted hypothesis tests to compare that semester's exam grades with the historic grades from past semesters. He found a significant difference. The difference was so significant that he was confident that the abnormally high grades were not due to random chance. Rather, the difference in exam grades was large enough to strongly suspect foul play. His suspicions were later confirmed by a student who tipped the professor off about students accessing the exam questions online. The guilty students eventually admitted to cheating and the entire class was required to take a new exam. While statistical analysis was not the only thing used to confirm the suspicion of cheating, the results of hypothesis testing triggered further inquiry.The real power of statistics is being able to use samples of data to infer something unknown about a larger population. We explored one aspect of this in Chapter 7 on confidence intervals. Here, we move onto using sample data to test hypotheses regarding the larger population. We start with the simplest case in which we use one sample of data to test a single numeric value regarding the population. Some good examples are hypothesis tests for quality control of products and services. Starbucks, for example, reports that a 16 oz cup of regular coffee (the grande size of Pike Place) has 310 milligrams (mgs) of caffeine.2 Of course, there is going to be some variation in the actual amount of caffeine found in every grande cup of coffee brewed across the United States and abroad. The caffeine content can depend on the water quality, how long the beans have been roasted, the age of the beans, the skillset of the barista, the type of filter, and many other things. Therefore, we would not expect every single 16 oz cup to have exactly 310 mg of coffee. Given this variation, it is possible to sample a number of cups of coffee to test whether the average
eBook - ePub- Deborah J. Rumsey(Author)
- 2019(Publication Date)
- For Dummies(Publisher)
Chapter 8Hypothesis Tests
IN THIS CHAPTERGeneral ideas for a hypothesis testType I and Type II errors in testingSpecific hypothesis tests for one or two population means or proportionsHypothesis testing is a statistician’s way of trying to confirm or deny a claim about a population using data from a sample. For example, you might read on the Internet that the average price of a home in your city is $150,000 and wonder if that number is true for the whole city. Or you hear that 65% of all Americans are in favor of a smoking ban in public places — is this a credible result? In this chapter, I give you the big picture of hypothesis testing as well the details for hypothesis tests for one or two means or proportions. And I examine possible errors that can occur in the process.Doing a Hypothesis Test
A hypothesis test is a statistical procedure that’s designed to test a claim. Typically, the claim is being made about a population parameter (one number that characterizes the entire population). Because parameters tend to be unknown quantities, everyone wants to make claims about what their values may be. For example, the claim that 25% (or 0.25) of all women have varicose veins is a claim about the proportion (that’s the parameter ) of all women (that’s the population ) who have varicose veins.Identifying what you’re testing
To get more specific, the varicose vein claim is that the parameter, the population proportion (p ), is equal to 0.25. (This claim is called the null hypothesis. ) If you’re out to test this claim, you’re questioning the claim and have a hypothesis of your own (called the research hypothesis, or alternative hypothesis
- Mikel J. Harry, Prem S. Mann, Ofelia C. De Hodgins, Richard L. Hulbert, Christopher J. Lacke(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
16 Single-Population Hypothesis Tests 16.1 OVERVIEW This chapter introduces the second topic in inferential statistics. tests of hypotheses. In a hypothesis test, we test a certain given theory or belief about a population parameter. We may want to find out, using some sample information, whether or not a given claim (or statement) about a population parameter is reasonable. This chapter discusses how to make such tests of hypotheses about the population mean µ and the population proportion p. As an example, a soft-drink company may claim that, on average, its cans contain 12 ounces (oz) of soda. A government agency may want to test whether or not such cans contain, on average, 12 oz of soda. As another example, according to the US Bureau of Labor Statistics, 57.3% of married women in the United States were working outside their homes in 1991. An economist may want to check if this percentage is still reasonable for this year. In the first of these two examples we are to test a hypothesis about the population mean µ, and in the second example we are to test a hypothesis about the population proportion p. 16.2 INTRODUCTION TO HYPOTHESIS TESTING Why do we need to perform a hypothesis test? Reconsider the example about soft-drink cans. Suppose that we take a sample of 100 cans of the soft drink under investigation. We then find out that the mean amount of soda in these 100 cans is 11.89 oz. On the basis of this result, can we state that, on average, all such cans contain less than 12 oz of soda and that the company is lying to the public? Not until we perform a hypothesis test can we make such an accusation. The reason is that the mean = 11. 89 oz is obtained from a sample. The difference between 12 oz (the required average for the population) and 11.89 oz (the observed average for the sample) may have occurred only because of the variability from can to can. Another sample of 100 cans may give us a mean of 12.04 oz
- Jay Devore, Nicholas Farnum, Jimmy Doi, , Jay Devore, Nicholas Farnum, Jimmy Doi(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
Hypothesis testing methods, as well as estimation methods, will be used extensively throughout the remainder of the book. 352 BSIP/UIG/Getty images Copyright 2013 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. 8.1 Hypotheses and Test Procedures 353 8.1 HYPOTHESES AND TEST PROCEDURES A statistical hypothesis, or just hypothesis, is a claim or assertion either about one or more population or process characteristics (parameters) or else about the form of the population or process distribution. Here are some examples of legitimate hypotheses: 1. Parameter: H9266 5 proportion of e-mail messages emanating from a certain system that are undeliverable Hypothesis: H9266 , .01 2. Parameters: H9262 1 5 true average lifetime for a particular name-brand tire (miles) H9262 2 5 true average lifetime for a less expensive store-brand tire Hypothesis: H9262 1 2 H9262 2 . 10,000 3. Parameters: H9266 1 5 proportion of individuals in a certain population with an AA genotype for a particular genetic characteristic H9266 2 5 proportion of individuals with an Aa genotype H9266 3 5 proportion of individuals with an aa genotype Hypothesis: H9266 1 5 .25, H9266 2 5 .50, H9266 3 5 .25 4. Population distribution: f ( x ), where x 5 the time between successive adjustments of a lathe process to correct for tool wear Hypothesis: x has an exponential distribution, that is, f 1 x 2 5 H9261 e 2 H9261 x for some H9261 . 0 In any hypothesis-testing problem, there are two competing hypotheses under con-sideration.
eBook - PDF- Ken Black, Ignacio Castillo, Amy Goldlist, Timothy Edmunds(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Testing Hypotheses about a Population Mean 355 example). In this section, we will summarize the process that we already learned how to apply and discuss some details and variations of the testing process that are specific to tests of the population mean. Assumptions As we described in Section 11.1, a formal hypothesis involves constructing the sampling dis- tribution of the test statistic. To perform a hypothesis test about the population mean, we need to construct the probability distribution of the sample mean. Recall from Chapter 9 that (by the Central Limit Theorem) the sampling distribution of the mean is well approximated by the normal distribution provided that the underlying distribution is normally distributed or that the sample size (n) is at least 30. As long as at least one of these conditions is satisfied for our hypothesis test, we can use the normal distribution to conduct the test. If we are not confident that the Central Limit Theorem applies to a problem (because the sample size is less than 30, and we are not confident that the population itself is normally distributed), then more advanced techniques are required to perform a hypothesis test of the mean. These techniques are beyond the scope of this textbook. Testing the Mean with the z Test ( σ Known) If the standard deviation of the population is known, then we can construct the sampling distribution of the mean as a normal distribution. We can then describe where any particular sample mean falls on that distribution by calculating its z score. That z score will be the test statistic for the hypothesis test. z Test for a Single Mean ( σ Known) µ σ = − z x n test This formula is valid when the standard deviation, σ , is known and if the sample size is large (n ≥ 30) for any population, or for small samples (n < 30) if x is known to be normally distributed in the population. (11.1) We call this value the “test statistic,” the “observed value,” or z test .
eBook - PDFBiostatistics
A Foundation for Analysis in the Health Sciences
- Wayne W. Daniel, Chad L. Cross(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
2. Assumptions. It is assumed that the sample comes from a population whose ages are approximately normally distributed. Let us also assume that the population has a known variance of 2 = 20. 3. Hypotheses. The hypothesis to be tested, or null hypothesis, is that the mean age of the population is equal to 30. The alternative hypothesis is that the mean age of the population is not equal to 30. Note that we are identifying with the alternative hypothesis the conclusion the researchers wish to reach, so that if the data permit rejection of the null hypothesis, the researchers’ conclusion will carry more weight, since the accompanying probability of reject- ing a true null hypothesis will be small. We will make sure of this by assigning a small value to , the probability of committing a type I error. We may present the relevant hypotheses in compact form as follows: H 0 ∶ = 30 H A ∶ ≠ 30 4. Test statistic. Since we are testing a hypothesis about a population mean, since we assume that the population is normally distributed, and since the population variance is known, our test statistic is given by Equation 7.2.1. 5. Distribution of test statistic. Based on our knowledge of sampling distributions and the normal distribution, we know that the test statistic is normally distributed with a mean of 0 and a variance of 1, if H 0 is true. There are many possible values of the test statistic that the present situation can generate; one for every possible sample of size 10 that can be drawn from the population. Since we draw only one sample, we have only one of these possible values on which to base a decision. 6. Decision rule. The decision rule tells us to reject H 0 if the computed value of the test statistic falls in the rejection region and to fail to reject H 0 if it falls in the nonrejection region. We must now specify the rejection and nonrejection regions.
- David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
However, if the sample results indicate H 0 cannot be rejected, the assumption that the manufacturer’s bottle filling operation is functioning properly cannot be rejected. In this case, no further action would be taken and the production operation would continue to run. The two preceding forms of the soft drink manufacturing hypothesis test show that the null and alternative hypotheses may vary depending upon the point of view of the re-searcher or decision maker. To correctly formulate hypotheses it is important to understand the context of the situation and structure the hypotheses to provide the information the researcher or decision maker wants. Summary of Forms for Null and Alternative Hypotheses The hypothesis tests in this chapter involve two population parameters: the population mean and the population proportion. Depending on the situation, hypothesis tests about a population parameter may take one of three forms: two use inequalities in the null hypo-thesis; the third uses an equality in the null hypothesis. For hypothesis tests involving a population mean, we let m 0 denote the hypothesized value and we must choose one of the following three forms for the hypothesis test. H 0 : H a : m $ m 0 m , m 0 H 0 : H a : m # m 0 m . m 0 H 0 : H a : m 5 m 0 m ± m 0 For reasons that will be clear later, the first two forms are called one-tailed tests. The third form is called a two-tailed test. In many situations, the choice of H 0 and H a is not obvious and judgment is necessary to select the proper form. However, as the preceding forms show, the equality part of the expression (either ≥ , ≤ , or = ) always appears in the null hypothesis. In selecting the proper form of H 0 and H a , keep in mind that the alternative hypothesis is often what the test is attempting to establish. Hence, asking whether the user is looking for evidence to support m < m 0 , m > m 0 , or m Þ m 0 will help determine H a .
- David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 420 Chapter 9 Hypothesis Tests a. Formulate hypotheses for a test to determine whether the sample data support the conclusion that the population annual expenditure for prescription drugs per person is lower in the Midwest than in the Northeast. b. What is the value of the test statistic? c. What is the p -value? d. At a 5 .01 , what is your conclusion? 21. Cost of Telephone Surveys. Fowle Marketing Research, Inc. bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less. If a longer mean survey time is necessary, a premium rate is charged. A sample of 35 surveys provided the survey times shown in the file Fowle . Based upon past studies, the population standard deviation is assumed known with s 5 4 minutes. Is the premium rate justified? a. Formulate the null and alternative hypotheses for this application. b. Compute the value of the test statistic. c. What is the p -value? d. At a 5 .01, what is your conclusion? 22. Time in Supermarket Checkout Lines. CCN and ActMedia provided a television channel targeted to individuals waiting in supermarket checkout lines. The channel showed news, short features, and advertisements. The length of the program was based on the assumption that the population mean time a shopper stands in a supermarket checkout line is 8 minutes. A sample of actual waiting times will be used to test this assumption and determine whether actual mean waiting time differs from this standard. a. Formulate the hypotheses for this application. b. A sample of 120 shoppers showed a sample mean waiting time of 8.4 minutes.
- Ken Black, Tiffany Bayley, Ignacio Castillo(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Other business scenarios resulting in hypothesis tests of a single mean might be as follows: • A financial investment firm wants to determine whether the average hourly change in the Toronto Stock Exchange over a 10-year period is +0.25. • A manufacturing company wants to determine whether the average thickness of a plastic bottle is 2.4 mm. • A retail store wants to determine whether the average age of its customers is less than 40 years. Formula 9.1 can be used to test hypotheses about a single population mean when σ is known if the sample size is large (n ≥ 30) for any population and for small samples (n < 30) if x is known to be normally distributed in the population. z Test for a Single Mean z = ¯ x − μ _____ σ ____ √ __ n (9.1) A survey of chartered professional accountants (CPAs) found that the average net income for sole proprietor CPAs is $74,914. 2 Because this survey is now more than 20 years old, an accounting analyst wants to test this figure by taking a random sample of 112 sole proprietor CPAs to determine whether the net income figure has changed. The analyst could use the eight steps of hypothesis testing to do so. Assume the population standard deviation of net incomes for sole proprietor CPAs is $14,530. Step 1 At Step 1, the hypotheses must be established. Because the analyst is testing to determine whether the figure has changed, the alternative hypothesis is that the mean net income is not $74,914. The null hypothesis is that the mean still equals $74,914. These hypotheses follow. H 0 : μ = $74,914 H a : μ ≠ $74,914 2 Adapted from Daniel J. Flaherty, Raymond A. Zimmerman, and Mary Ann Murray, “Benchmarking against the Best,” Journal of Accountancy (July 1995): 85–88. 292 CHAPTER 9 Statistical Inference: Hypothesis Testing for Single Populations Step 2 Step 2 is to determine the appropriate statistical test and sampling distribution.
No longer available |Learn more- Jessica Utts, Robert Heckard(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Bold exercises have answers in the back of the text. For all exercises requiring hypothesis tests, make sure that you check conditions, define all parameters, and state your conclu-sion in words in the context of the problem. Section 13.1 Skillbuilder Exercises 13.1 Explain whether each of the following statements is true. a. One of the two possible conclusions in hypothesis testing is to accept the null hypothesis. b. The statements “reject the null hypothesis” and “accept the alternative hypothesis” are equivalent. 13.2 Explain whether each of the following statements is true. a. Hypotheses and conclusions from hypothesis testing apply only to the samples on which they are based. b. The p -value is calculated with the assumption that the null hypothesis is true. 13.3 For each of the following research questions, specify whether the parameter of interest is one population mean, the popu-lation mean of paired differences, or the difference between the means of two populations. a. Nutrition trends have changed over the years, and this may affect growth. Researchers want to know whether Copyright 2014 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. Testing Hypotheses about Means 551 Dataset available but not required Bold exercises answered in the back the mean height of 25-year-old women is the same as the mean height of 45-year-old women. b. You plan to fly from New York to Chicago and have a choice of two flights. You are able to find out how many minutes late each flight was for a random sample of 25 days over the past few years.
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