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21-1 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 21 Hypothesis Testing Introductory Mathematics & Statistics for Business

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21-2 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Learning Objectives Understand the principles of statistical inference Formulate null and alternative hypotheses Understand one-tailed and two-tailed tests Understand type I and type II errors Understand test statistics Understand the significance level of a test Understand and calculate critical values Understand the regions of acceptance and rejection Calculate and interpret a one-sample z-test statistic Calculate and interpret a one-sample t-test statistic Calculate and interpret a paired t-test statistic Calculate and interpret a two-sample t-test statistic Understand and calculate p-values

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21-3 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.1 Statistical inference One of the major roles of statisticians in practice is to draw conclusions from a set of data This process is known as statistical inference We can put a probability on whether a conclusion is correct within reasonable doubt The major question to be answered is whether any difference between samples, or between a sample and a population, has occurred simply as a result of natural variation or because of a real difference between the two

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21-4 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.1 Statistical inference (cont…) In this decision-making process we usually undertake a series of steps, which can include the following: 1. collecting the data 2. summarising the data 3. setting up an hypothesis (i.e. a claim or theory), which is to be tested 4. calculating the probability of obtaining a sample such as the one we have if the hypothesis is true 5. either accepting or rejecting the hypothesis

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21-5 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.1 Statistical inference (cont…) The null hypothesis –A technique for dealing with these problems begins with the formulation of an hypothesis –The null hypothesis is a statement that nothing unusual has occurred. The notation is H o –The alternative hypothesis states that something unusual has occurred. The notation is H 1 or H A –Together they may be written in the form: H o : (statement) v H 1 (alternative statement) (where v stands for versus)

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21-6 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.1 Statistical inference (cont…) The null hypothesis (cont…) –E.g. Question: Is the population mean, μ, equal to a specified value? Hypotheses: H 0 : The population mean, μ, is equal to the specified value. v. H 1 : The population mean, μ, is not equal to the specified value –Another way of expressing the null and alternative hypotheses is in the form of symbols, e.g.

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21-7 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.1 Statistical inference (cont…) The alternative hypothesis –The alternative hypothesis may be classified as two-tailed or one-tailed –Two-tailed test (two-sided alternative) we do the test with no preconceived notion that the true value of μ is either above or below the hypothesised value of μ 0 the alternative hypothesis is written: H 1 : µ µ o

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21-8 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.1 Statistical inference (cont…) Significance level –After the appropriate hypotheses have been formulated, we must decide upon the significance level (or –level) of the test –This level represents the borderline probability between whether an event (or sample) has occurred by chance or whether an unusual event has taken place –The most common significance level used is 0.05, commonly written as = 0.05 –A 5% significance level says in effect that an event that occurs less than 5% of the time is considered unusual

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21-9 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.2 Errors There are two possible errors in making a conclusion about a null hypothesis –Type I errors occur when you reject H 0 (i.e. conclude that it is false) when H 0 is really true. The probability of making a type I error is, in fact, equal to, the significance level of the test –Type II errors occur when you accept H 0 (i.e. conclude that it is true) when H 0 is really false. The probability of making a type II error is denoted by the Greek letter (beta)

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21-10 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.3 The one-sample z-test Deals with the case of a single sample being chosen from a population and the question of whether that particular sample might be consistent with the rest of the population To answer this question, we construct a test statistic according to a particular formula Is important that the correct test be used, since the use of an incorrect test statistic can lead to an erroneous conclusion

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21-11 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.3 The one-sample z-test (cont…) Information required in calculation: –the size (n) of the sample –the mean of the sample –the standard deviation (s) of the sample Other information of interest might include: normal distribution 1. Does the population have a normal distribution? standard deviation 2. Is the populations standard deviation known? 3. Is the sample size (n) large?

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21-12 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.3 The one-sample z-test (cont…) There are different cases for the one-sample z-test statistic Case I is a situation where: 1. the population has a normal distribution and 2. the population standard deviation,, is known Case II is a situation where: 1. the population has any distribution 2. the sample size, n, is large (i.e. at least 25), and 3. the value of is known In both these cases we can use a z-test statistic defined by:

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21-13 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.3 The one-sample z-test (cont…) Case III is a situation where: 1. the population has any distribution 2. the sample size, n, is large (i.e. at least 25), and 3. the value of is unknown (however, since n is large, the value of is approximated by the sample standard deviation, s) In this case we can use a z-test statistic defined by:

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21-14 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.3 The one-sample z-test (cont…) A critical value is one that represents the cut-off point for z-test statistics in deciding whether we do not reject or do reject H 0. The particular critical value to use depends on two things: 1. whether we are using a one-sided or two-sided test, and 2. the significance level used

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21-15 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.3 The one-sample z-test (cont…) Rules for not rejecting or rejecting H 0 in a one- sample z-test

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21-16 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.4 The one-sample t-test Consider a test that has the same objective as a one-sample z- test, that is to determine whether a sample is significantly different from the population as a whole However, the one-sample t-test has a different set of assumptions: Case IV is a situation where: 1. the population is normally distributed 2. the population standard deviation,, is unknown and 3. the sample size, n, is small Then we can use a t-test statistic defined as:

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21-17 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.4 The one-sample t-test (cont…) Unlike the z-test statistic, the t-test statistic has associated with it a quantity called degrees of freedom The degrees of freedom are denoted by the Greek letter υ and are defined by υ = n – 1. Degrees of freedom relate to the fact that the sum of all deviations in a sample of size n must add up to zero When using a t-test statistic, we must use special critical values in a table This information is given in Table 7, and the critical values are given to three decimal places After the value of the t-test statistic is found, the method of either not rejecting or rejecting H 0 is similar to Rules (1) to (6) described for the z-test

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21-18 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.5 Drawing a conclusion Once a decision has been made to either not reject or reject H 0, a conclusion should be written in terms of what the actual problem is To summarise, the steps that should be undertaken to perform a one-sample test are: 1. Set up the null and alternative hypotheses. This includes deciding whether you are using a one-sided or two-sided test 2. Decide on the significance level 3. Write down the relevant data 4. Decide on the test statistic to be used 5. Calculate the value of the test statistic 6. Find the relevant critical value and decide whether H 0 is to be not rejected or rejected 7. Draw an appropriate conclusion

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21-19 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.6 The paired t-test Now we will consider problems in which there are two samples that are to be compared with each other These are often referred to as two-sample problems and there are a number of test statistics available In some instances, the two samples have a structure such that the data are paired A comparison of the values in two samples requires the calculation of a two-sample test statistic In such cases, the most commonly used test statistic is a paired t-test, which takes into account this natural pairing

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21-20 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.6 The paired t-test (cont…) The steps involved in the analysis are as follows: 1. Construct the null and alternative (two-tailed) hypotheses. H 0 always states that there is no difference between the two samples. H 1 always states that there is some difference between the two samples. In symbol form these can be written as: H 0 : μ d = 0 v.H 1 : μ d 0 2. Calculate the actual differences between the values in the two samples. It is important to retain the minus sign if the subtraction yields a negative number 3. Calculate the mean and standard deviation of the values of the differences

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21-21 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.6 The paired t-test (cont…) 4. Calculate the value of the paired t-test statistic using: where n = the number of pairs of differences μ d = 0 (according to the null hypothesis) and with = n – 1 degrees of freedom

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21-22 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.6 The paired t-test (cont…) 5. Look up the critical value in Table 7 for the desired significance level If | test statistic | > critical value, we reject H 0 and the conclusion is that H 1 applies If | test statistic | < critical value, we cannot reject H 0 and the conclusion is that there is no evidence of a significant difference between the two samples

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21-23 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.7 The two-sample t-test This time, suppose that the samples are not paired; that is, they are independent The two samples need not contain the same number of observations The most common test statistic used in this situation is a two-sample t-test (also known as a pooled t-test) The steps for using a two-sample t-test are as follows: 1. Construct the null and alternative (two-tailed) hypotheses. H 0 always states that there is no difference between the two samples, while H 1 always states that there is some difference between the two samples H 0 : μ 1 = μ 2 vH 1 : μ 1 μ 2

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21-24 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.7 The two-sample t-test (cont…) 2. Calculate the following statistics from the samples: the number of observations in Sample 1 and 2, the mean of Sample 1 and 2, the standard deviation of Sample 1 and 2 3. Calculate the value of the pooled standard deviation 4. Calculate the value of the two-sample t-test statistic

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21-25 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.7 The two-sample t-test (cont…) 5. Look up the critical value in Table 7 for the desired significance level. If | test statistic | > critical value, we reject H 0 and the conclusion is that H 1 applies. Hence, there is a significant difference between the two samples If | test statistic | < critical value, we cannot reject H 0 and the conclusion is that there is no evidence of a significant difference between the two samples

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21-26 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 21.8 p-values An alternative to using critical values for testing hypotheses is to use a p-value approach The value of the test statistic is still calculated in the usual way In this case we now calculate the probability of obtaining a value as extreme as the value of the test statistic if H 0 were true Compare the p-value with. Then: If p-value >, we cannot reject H 0 at that -level If p-value <, we reject H 0 at that -level

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21-27 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Summary In this chapter we looked at the principles of statistical inference We formulated null and alternative hypotheses We understood –one-tailed and two-tailed tests –type I and type II errors –test statistics –the significance level of a test –the regions of acceptance and rejection We understood and calculated critical values We calculated and interpreted –a one-sample z-test statistic –a one-sample t-test statistic –a paired t-test statistic –a two-sample t-test statistic We understood and calculated p-values

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