Binomial Hypothesis Test
A binomial hypothesis test is a statistical method used to determine if the proportion of successes in a sample is significantly different from a hypothesized proportion. It involves calculating the probability of observing the sample results or more extreme results, assuming the null hypothesis is true. The test is commonly used in various fields to make inferences about proportions and probabilities.
8 Key excerpts on "Binomial Hypothesis Test"
eBook - ePub- Richard Startup, Elwyn T. Whittaker(Authors)
- 2021(Publication Date)
- Routledge(Publisher)
...rejecting a valid null hypothesis, even at the risk of failing to claim a positive result which is, in fact, true. What this amounts to is that the chance of making a type I error is kept to a low magnitude. The investigator does this by setting a relatively small level of significance α, for this is the probability of a type I error. An α of 0.05 or 0.01 (or even 0.001) is customary. If, for instance, a 0.05 (or 5 per cent) significance level is chosen in designing a test of a null hypothesis, then there are 5 chances in 100 that we would reject the hypothesis were it valid. The Binomial Test Our discussion so far of hypothesis testing has been somewhat abstract and the reader may at first find this area of statistics a little daunting. However, the logic of the method can be clarified by the presentation of a straightforward example which makes use of the binomial distribution (see p. 60). We shall return to the already-presented data on textile workers shortly, but let us first modify the problem by supposing that a much smaller sample has been selected. Example. 40.0 per cent of adults possess educational paper qualifications, but among a random sample of ten textile workers only one does. It is expected that the proportion of textile workers with these qualifications will be less than 40.0 per cent, but do these figures provide firm evidence that the proportion in this occupational group is indeed less than that of adults as a whole? If we let p represent the proportion of textile workers in the region who possess qualifications and q the proportion who do not, then the null hypothesis H 0 is that p = 0.40 (and hence q = 0.60). On the other hand, the alternative hypothesis H 1, which is directional, is that p < 0.40 (and hence q > 0.60). Let us test the null hypothesis using the significance level α = 0.05. In the sample one textile worker is qualified...
- Michael R. Chernick(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...But it would be easily detected by a paired t- test or a nonparametric approach (sign test). 6.6 TESTING A SINGLE BINOMIAL PROPORTION The binomial distribution depends on two parameters n and p. It represents the sum of n independent Bernoulli trials. A Bernoulli trial is a test with two possible outcomes that are often labeled as success and failures. The binomial random variable is the total number of successes out of the n trials. So the binomial random variable can take on any value between 0 and n. The binomial distribution has mean equal to np and variance np (1 − p). These results are to construct the pivotal quantity for construction confidence interval and testing hypotheses about the unknown parameter p. For a confidence interval, the central limit theorem can be applied for large n. So let, where is the sample estimate of p, and X is the number of successes. The estimate we use is = X/n. Z has an approximate normal distribution with mean 0 and variance 1. This is the continuity-corrected version. Removing the term −1/2 from the numerator gives an approximation without the continuity correction. Here Z can be used to invert to make a confidence interval statement about p using the standard normal distribution. However, for hypothesis testing, we can take advantage of the fact that p = p 0 under the null hypothesis to construct a more powerful test. p 0 is used in place of and p in the definition of Z. So we have. Under the null hypothesis, this continuity-corrected version has an approximate standard normal distribution. 6.7 RELATIONSHIP BETWEEN CONFIDENCE INTERVALS AND HYPOTHESIS TESTS Suppose we want to test the null hypothesis that μ 1 − μ 2 = 0 versus the two-sided alternative that μ 1 − μ 2 ≠ 0. We wish to test at the 0.05 significance level. Construct a 95% confidence interval for the mean difference. For the hypothesis test, we reject the null hypothesis if and only if the confidence interval does not contain 0...
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...We described this process, in detail, here to emphasise how it is applied to hypothesis testing. In the next chapter, we will see how we can make use of the normal distribution as an approximation of the binomial distribution. This approach simplifies the computation of probabilities. We will also show how to apply the normal distribution in testing hypotheses that would usually require the need of the binomial distribution. 5.10 The sign test A further application of the binomial distribution in hypothesis testing occurs in the sign test, a very simple, but extremely useful inferential test. What follows is an example of how the sign test can be used to test a hypothesis. A sample of 12 subjects is randomly selected and each subject is asked to commit to memory a series of 24 randomly presented words, where 12 of these words do not have any association between them, while the remaining 12 belong to the same category (e.g., pieces of furniture). At test subjects are asked to recall all the words they remember in any order they want, and the number of categorised and non-categorised words recalled by each subject is recorded. The researcher aims to assess whether or not categorised words are more easily recalled than non-categorised words. A set of fictitious data is presented in Table 5.3. The question to answer is then: Does recall of categorised words differ from recall of non-categorised words? Table 5.3 (a) Individual performances in the recall of lists of categorised and non-categorised words (b) Binomial distribution of r successes out of 12 trials, with P(Success) = 0.5 In applying the sign test, the first thing to do is to obtain the sign of the difference between each pair of performances. In our case, we could take the difference between the number of categorised words being recalled minus the number of non-categorised words recalled...
eBook - ePubSensory Evaluation of Food
Statistical Methods and Procedures
- Michael O'Mahony(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...So a statistical test will do one of two things. If the probability of getting our result is low, should H 0 be true, we reject H 0 and say there is a difference. If the probability of getting our result is high, should H 0 be true, we do not then reject H 0 ; we say that our data are insufficient to show a difference. We do not accept H 0 ; we merely do not reject it. Consider another example. The null hypothesis may be a hypothesis that states that there is no correlation between two sets of numbers. We perform our statistical test to see whether the observed correspondence in our sample of numbers indicates a correlation in their respective populations. We calculate the probability of getting this degree of correlation should H 0 (no correlation) be true. If the probability of getting our result is low were H 0 true, we reject H 0 and accept the alternative hypothesis (H 1) that there is a correlation. Should the probability be high, we do not reject H 0. We say that our data are insufficient to indicate a correlation in the two respective populations. Note again that we do not accept H 0 ; we just do not reject it. This covers the whole logic of statistics. Whatever the test, we set up a null hypothesis. Should the probability of getting our result on H 0 be low, we reject it; if not, we do not reject it. Statistical tests are designed simply to calculate such probabilities. You do not have to calculate these probabilities yourself; it has all been done—you merely use a set of tables. Applying the Binomial Expansion And now back to our example. H 0 states that there are no differences in the numbers of men and women in the university...
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 - ePubStatistics
The Essentials for Research
- Henry E. Klugh(Author)
- 2013(Publication Date)
- Psychology Press(Publisher)
...10 Chi Square The binomial distribution and its normal approximation provide a test of significance for hypotheses about dichotomous data. It is an appropriate test to use when observations can be classified into one of two possible categories such as “yes-no,” or “male-female,” or “correct-incorrect.” When data can be classified in more than two categories, the binomial no longer provides a test of significance. For example, a response might be classified as occurring “always,” “often,” “sometimes,” or “never.” In that situation, when the population is composed of more than two classes of events, it would appear reasonable to employ the multinomial distribution to determine the probability of obtaining particular kinds of samples. This is theoretically possible but the calculations required quickly become prohibitive. Consequently, we use a distribution that approximates the multinomial (and the binomial) distribution. This distribution, and the test of significance named for it, is called Chi Square (χ 2). 10.1 The Chi Square Distribution The chi square distribution differs in some important ways from the binomial and normal distributions we have already discussed, but no new principles are involved in chi square’s use as a test of significance. Before discussing chi square, let us briefly review the normal approximation of the binomial as a test of significance. We begin with some hypothesis about the proportion of events in a binomial population which, given the sample size, allows us to calculate the mean and standard error of a theoretical sampling distribution. With this information we can calculate the z equivalent of any obtained sample proportion. When it can be assumed that z is normally distributed, Table N gives the probability of obtaining a value of z as large or larger than that yielded by the sample...
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- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...13 Hypothesis Testing, Parametric Tests, Distribution Tests, and Tests of Significance 13.1 Introduction To help decision makers decide about a population, the data contained in a sample from that population is examined. To make a decision regarding the population parameter based on the sample information, we are supposed to make an assumption about the population parameters. The assumption made about the population is referred to as a ‘hypothesis’. This assumption may be true or false. The methodology that helps to conclude whether the assumption made is true is called ‘hypothesis testing’. It can be classified into a null hypothesis (H 0) and an alternative hypothesis (H 1). 13.2 Null Hypothesis (H 0) According to R. A. Fisher a null hypothesis can be defined as ‘the hypothesis that is tested for possible rejection under the assumption that it is true’. In other words, H 0 asserts that there is no significant difference between the value of the population parameter being tested with the value of the statistic evaluated from a sample drawn. The null hypothesis normally specifies one of the parameters of the population of interest; the term ‘null hypothesis’ reflects the idea that this is a hypothesis of no difference. Hence, H 0 always includes a statement of equality. 13.3 Alternative Hypothesis (H 1) H 1 refers to the alternative available when the null hypothesis must be rejected. Let us assume a situation in which you need to test a hypothesis about a population...
