Hypothesis Test of Two Population Proportions
The hypothesis test of two population proportions is a statistical method used to compare the proportions of two different populations. It involves formulating null and alternative hypotheses, calculating the test statistic, and determining the p-value to make inferences about whether the proportions are significantly different. This test is commonly used in research and decision-making processes to assess the significance of differences between population proportions.
8 Key excerpts on "Hypothesis Test of Two Population Proportions"
- Mikel J. Harry, Prem S. Mann, Ofelia C. De Hodgins, Richard L. Hulbert, Christopher J. Lacke(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
...Assuming that the null hypothesis is true and p 1 = p 2, a common value of and, denoted by, is calculated by using one of the following formulas: Which of these formulas is used depends on whether the values of x 1 and x 2 or the values of and are known. Note that x 1 and x 2 are the number of elements in each of the two samples that possess a certain characteristic. This value of p c is called the pooled sample proportion. Using the value of the pooled sample proportion, we compute the standard error of, under the assumption p 1 = p 2, as The value of the test statistic z for is calculated as The value of p 1 – p 2, which is typically zero, is given in the null hypothesis. Examples 17.15 and 17.16 illustrate the procedure to test hypotheses about the difference between two population proportions for large samples. Example 17.15 Reconsider Example 17.14 regarding the percentages of users of two toothpastes who will never switch to another toothpaste. At the 1% significance level, can we conclude that the proportion of users of toothpaste A who will never switch to another toothpaste is greater than the proportion of users of toothpaste B who will never switch to another toothpaste? Solution Let p 1 and p 2 be the proportions of all users of toothpastes A and B, respectively, who will never switch to another toothpaste and let and be the corresponding sample proportions. The given information is Step 1. Select the type of test to use and check the underlying conditions. As we discussed in Example 17.14, these data correspond to two independent samples. Furthermore, we showed that the sample sizes are large enough to use the normal distribution for performing the test. Step 2. State the null and alternative hypotheses. Note that we are testing to find whether the proportion of users of toothpaste A who will never switch to another toothpaste is greater than the proportion of users of toothpaste B who will never switch to another toothpaste...
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...10 Hypothesis Testing for Single Population Mean and Proportion Hypothesis testing is used to discover whether there is enough statistical evidence in favor of or against a belief about a population parameter. Testing is used to infer results from sample data as they apply to the overall population. In this chapter, we will demonstrate how to use the concept of hypothesis testing to make a smart decision. We will also dwell on its applications in business. 10.1 Null and Alternative Hypotheses A hypothesis is a statement speculating upon the result of a research study which can be used to describe a population parameter. The hypothesis indicating no association among groups or between measured attributes is called the null hypothesis ; this is the hypothesis under investigation we are trying to disprove. It is denoted H 0. However, the hypothesis that observations represent the real effect of the study is called the alternative hypothesis. An alternative hypothesis is denoted H 1. For example, if the government wants to know if the unemployment rate in the country is different from the 5 percent claimed by the National Bureau of Statistics, the null hypothesis for this scenario is H 0 : μ = 5 % versus the alternative hypothesis H 1 : μ ≠ 5 %. 10.2 Type I and Type II Error A type I error occurs when the null hypothesis is rejected when it is true. This error is also known as a false positive. The probability of rejecting a null hypothesis when it is true is denoted by α —i.e., P(rejecting H 0 | H 0 is true) = α. Type I error is sometimes called a producer’s risk or a false alarm. This error is usually set by the researcher—the lower the α value, the lower of chance of committing type I error. The probability value (p -value) is often set to be 0.05, except in biomedical research where the p -value is set to be 0.01 because it deals with human life...
eBook - ePub- Andrew F. Hayes(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...It is not clear from my description above whether or not there has was any attempt to randomly sample from some population of people taking a public speaking course, and it isn’t at all clear just how we could ever randomly sample from such a population. I merely described that 20 students provided data, but I did not specify how those students were obtained. As I discuss at various points throughout this book, random sampling isn’t always feasible or even desirable. Whether or not you should randomly sample depends on the kinds of inferences you are interested in making. So hypothesis testing plays an important role in the evaluation of research findings, but its role is limited to evaluating the data available. It is up to the researcher to maximize the quality of the data through thoughtful research design strategies, such that when the p -value is computed, the substantive interpretation and conclusion about what the data are telling you can be made unequivocally. 8.2 Testing a Hypothesis About a Population Proportion To this point I have been speaking in abstractions, and I’d now like to formalize the procedure by showing the details for testing a hypothesis, including how to compute the p -value when testing a hypothesis commonly of interest to communication researchers: a hypothesis about the value of a population proportion. 8.2.1 Testing a Nondirectional (“Two-tailed”) Hypothesis Suppose you work on the campaign staff for a city council member who is up for reelection, and you have been assigned to poll the council member’s constituents to see if the constituents have a preference for raising some needed money for the city through an income tax increase or an increase in the city sales tax...
eBook - ePubBiostatistical Design and Analysis Using R
A Practical Guide
- Murray Logan(Author)
- 2011(Publication Date)
- Wiley-Blackwell(Publisher)
...Logically however, theories (and thus hypothesis) cannot be proved, only disproved (falsification) and thus a null hypothesis (H 0) is formulated to represent all possibilities except the hypothesized prediction. For example, if the hypothesis is that there is a difference between (or relationship among) populations, then the null hypothesis is that there is no difference or relationship (effect). Evidence against the null hypothesis thereby provides evidence that the hypothesis is likely to be true. The next step in hypothesis testing is to decide on an appropriate statistic that describes the nature of population estimates in the context of the null hypothesis taking into account the precision of estimates. For example, if the null hypothesis is that the mean of one population is different to the mean of another population, the null hypothesis is that the population means are equal. The null hypothesis can therefore be represented mathematically as: H 0 : μ 1 = μ 2 or equivalently: H 0 : μ 1 – μ 2 = 0. The appropriate test statistic for such a null hypothesis is a t -statistic: where (1 – 2) is the degree of difference between sample means of population 1 and 2 and expresses the level of precision in the difference. If the null hypothesis is true and the two populations have identical means, we might expect that the means of samples collected from the two populations would be similar and thus the difference in means would be close to 0, as would the value of the t -statistic. Since populations and thus samples are variable, it is unlikely that two samples will have identical means, even if they are collected from identical populations (or the same population). Therefore, if the two populations were repeatedly sampled (with comparable collection technique and sample size) and t -statistics calculated, it would be expected that 50% of the time, the mean of sample 1 would be greater than that of population 2 and visa versa...
eBook - ePubStats Means Business
Statistics and Business Analytics for Business, Hospitality and Tourism
- John Buglear(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...The test statistic is: t = 7.75 − 8.00 3.05 √ 12 = − 0.25 0.88 = − 0.28 to two places From Table A.4 on pages 282–283, t 0.10,11 is 1.363. The alternative hypothesis is that the population mean salary difference is less than £8000, so the critical value is −1.363. A sample mean that produces a test statistic less than this would lead us to reject the null hypothesis. In this case, although the sample mean is less than £8000, the test statistic, −0.28, is not less than the critical value and the null hypothesis cannot be rejected. The population mean of the salary differences could well be £8000. 8.6 Testing hypotheses about population proportions In many respects, the procedure we use to test hypotheses about population proportions is similar to the way we test other types of hypothesis. We begin with a null hypothesis that specifies a population proportion to be tested, which we represent by the symbol π 0 (the Greek letter ‘pi’ is the symbol for the population proportion). If the null hypothesis is one of the ‘equal to’ types, we conduct a two-tail test. If it is ‘less than’ or ‘greater than’, we conduct a one-tail test. The three possible combinations of hypotheses are listed in Table 8.5. Table 8.5 Types of hypotheses for the population proportion Null Hypothesis Alternative Hypothesis Type of test H 0 : π = π 0 H 1 : π ≠ π 0 (‘not equal’) Two-sided H 0 : π ≤ π 0 H 0 : π > π 0 (‘greater than’) One-sided H 0 : π ≥ π 0 H 0 : π < π 0 (‘less than’) One-sided In this table, π 0 represents the value of the population proportion that is to be tested. We calculate the test statistic from the sample proportion, represented by the symbol p, which comes from the sample data that we want to use to test the hypothesis. We assume that the sample proportion belongs to a sampling distribution that has a mean of π 0 and a standard error of: π 0 (1 − π 0) n Notice that we use the proportion from the null hypothesis to calculate the standard error, not the sample proportion...
eBook - ePubMedical Statistics
A Textbook for the Health Sciences
- Stephen J. Walters, Michael J. Campbell, David Machin(Authors)
- 2020(Publication Date)
- Wiley-Blackwell(Publisher)
...6 Hypothesis Testing, P ‐values and Statistical Inference 6.1 Introduction 6.2 The Null Hypothesis 6.3 The Main Steps in Hypothesis Testing 6.4 Using Your P-value to Make a Decision About Whether to Reject, or Not Reject, Your Null Hypothesis 6.5 Statistical Power 6.6 One-sided and Two-sided Tests 6.7 Confidence Intervals (CIs) 6.8 Large Sample Tests for Two Independent Means or Proportions 6.9 Issues with P-values 6.10 Points When Reading the Literature 6.11 Exercises Summary The main aim of statistical analysis is to use the information gained from a sample of individuals to make inferences or form judgements about the parameters (e.g. the mean) of a population of interest. This chapter will discuss two of the basic approaches to statistical analysis: estimation (with confidence intervals (CIs)) and hypothesis testing (with P‐ values). The concepts of the null hypothesis, statistical significance, the use of statistical tests, P‐ values and their relationship to CIs are introduced. The difficulties with the use and mis‐interpretation of P ‐values are discussed. 6.1 Introduction We have seen that, in sampling from a population which can be assumed to have a Normal distribution, the sample mean can be regarded as estimating the corresponding population mean μ. Similarly, s 2 estimates the population variance, σ 2. We therefore describe the distribution of the population with the information given by the sample statistics and s 2. More generally, in comparing two populations, perhaps the population of subjects exposed to a particular hazard and the population of those who were not, two samples are taken, and their respective summary statistics calculated...
eBook - ePub- Thomas Pfaff(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...13 Testing Proportions In testing a single proportion we present two options. The first is the traditional z-test, prop.test, and the second is an exact test, binom.test. The prop.test generalizes to two or more populations and we provide an example of testing two proportions. Our examples here will use data generated from a binomial distribution using rbinom. We end the chapter with an example of the error associated with the choice of α. 13.1 Tests and Intervals for One and Two Proportions We begin this section by generating data to illustrate testing proportions. We let heads be the number of successes of a binomial random variable with n=123 and a probability of successes of 0.5. In this case, we have 66 successes and results will vary. We will test to see if this outcome is distinguishable from a probability of success of 0.55 in two ways. We will use a z-test and an exact test. R Code > heads=rbinom(1,size=123,prob=0.5) > heads [1] 66 The function prop.test performs a one-proportion z-test. We provide the number of successes, stored in heads, the sample size, 123, and a value for p 0, 0.55. The alternative hypothesis is set by alternative with the options two.sided, less, or greater placed in quotes, where two.sided is the default. The default confidence interval is 95%, and can be set to, say, 90% with conf.level=0.90. By default correct=TRUE so that the continuity correction is used. It can be set to FALSE. Our p-value is 0.8349, so we fail to reject the null hypothesis. R Code > result.z=prop.test(heads,123,p=0.55, alternative="two.sided") > result.z 1-sample proportions test with continuity correction data: heads out of 123, null probability 0.55 X-squared = 0.043443, df = 1, p-value = 0.8349 alternative hypothesis: true p is not equal to 0.55 95 percent confidence interval: 0.4447004 0.6261536 sample estimates: p 0.5365854 Alternatively, we can perform an exact test by using binom.test...
eBook - ePub- Derek L. Waller(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...The sample or test value of the Student’s t remains unchanged at 1.4498. Now, 1.4498 > 1.3562 and thus we reject the null hypothesis and conclude that the publicity for the program is correct and that the average weight loss is greater than 10 kg. Using the p -value approach for this test then with function TDIST (Excel 2007) or T.DIST.RT (Excel 2010 or later) in Excel for a one-tailed test, the area in the tail is still 8.64%. This is less than 10.00% and so our conclusion is the same in that we reject the null hypothesis. Figure 9.9 shows the concepts for the p -value and the Student’s t distributions. Figure 9.9 Health spa and weight loss (10% significance). Again, as in all hypothesis testing, remember that the conclusions are sensitive to the level of significance used in the test. Differences between the proportions of two populations with large samples There are situations where we might be interested to know if there is a significant difference between the proportion or percentage of some criterion of two different populations. For example, is there a significant difference between the percentage output of two of a firm’s production sites? Is there a difference between the health of British people and Americans? (The answer is yes, according to a study in the Journal of the American Medical Association. 3) Is there a significant difference between the percentage effectiveness of one drug and another drug for the same ailment? In these situations we take samples from each of the two groups and test for the percentage or proportional difference in the two populations...
