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CHAPTER 10 HYPOTHESIS TESTING Outline • Hypothesis testing – The context and some terms – Testing population mean when the population variance is known • Two-tail test • One-tail test – The p-value of a test of hypothesis – The probability of a Type II error 1 HYPOTHESIS TESTING THE CONTEXT • Example 1: A supervisor of a production line wants to determine if the production time of a critical part is the same as its design time, say 100 seconds. A random sample of parts is taken and their production times are measured. Does the sample information provide enough evidence that the production time of the part is 100 seconds? Of course, both of the following are important – The production times sampled and – The size of the sample • In the above context, hypothesis testing provides a technique to conclude if the production time of the part is 100 seconds. 2 HYPOTHESIS TESTING THE CONTEXT • Example 2: Suppose that a manager wants to produce a new product if more than 10% potential customers buy the product. A random sample of potential customers is asked whether they would buy the product. Does the sample information provide enough evidence that more than 10% potential customers will buy the new product? Of course, both of the following are important – The yes/no answers provided by the respondents and – The size of the sample • In the above context, hypothesis testing provides a technique to conclude if more than 10% potential customers will buy the new product. 3 HYPOTHESIS TESTING THE CONTEXT • Example 3: Suppose that a quality control inspector wants to determine if less than 2% items are defective. A random sample of items are checked and inspected. Does the sample information provide enough evidence that less than 2% items are defective? Of course, both of the following are important – The proportion defectives observed in the sample and – The size of the sample • In the above context, hypothesis testing provides a technique to conclude if less than 2% items are defective. 4 HYPOTHESIS TESTING SOME TERMS • Null Hypothesis, HO – The null hypothesis always specifies a single value. For example, suppose that it is required to determine if the population mean is 10. Then, the null hypothesis is H O : 100 – Note that since the null hypothesis always specifies a single value, none of the below may be a null hypothesis H O : 100 H O : 100 H O : 100 H O : 80 120 5 HYPOTHESIS TESTING SOME TERMS • Alternative Hypothesis, HA – The alternative hypothesis is very important because the conclusion of the hypothesis testing is stated in terms of alternative hypothesis – Hypothesis testing provides a technique to determine if there is enough statistical evidence that the alternative hypothesis is true. – There are three forms of alternative hypothesis: Two-tail test One-tail (right-tail) test One-tail (left-tail) test H A : p 0.02 H O : 100 H O : p 0.10 H A : 100 H A : p 0.10 H A : p 0.02 Example 1 Example 2 Example 3 6 HYPOTHESIS TESTING SOME TERMS • Alternative Hypothesis, HA (One- and Two-Tail Tests) – It is very important to choose the right form of the alternative hypothesis. The form depends on the context. – In Example I, the supervisor wants to know if the mean is 100 or different from 100. Both the too large and too small values are equally undesirable. It is appropriate to reject the claim if the sample mean is much different from 100. So, the most appropriate test is the two-tail test. – In Examples 2 and 3 too small and too large observations do not lead to the same action. When this happens, a one-tail test is used. 7 HYPOTHESIS TESTING SOME TERMS • Alternative Hypothesis, HA (Left- and Right-Tail Tests) – Choose between the left- and right-tail tests carefully. – In Example 2 the manager wants to know if the proportion is more than 0.10. So, H A : p 0.10 and the most appropriate test is a right-tail test. – In Example 3 the inspector wants to know if the proportion is less than 0.02. So, H A : p 0.02 and the most appropriate test is a left-tail test. 8 HYPOTHESIS TESTING SOME TERMS • Test Statistic and Rejection Region – The test statistic is computed from the sample data. – The test statistic is different for different tests. Only z-test is discussed in Chapter 10 and the test statistic for the ztest is x z / n – The test statistic is the same for both one-tail and twotail tests. The rejection regions for one-tail and two-tail tests are different. – If the test statistic lies in the rejection region, the null hypothesis is rejected, else the null hypothesis is not rejected (Beware: not rejected accepted) 9 HYPOTHESIS TESTING SOME TERMS • Rejection Region and Level of Significance, – Conclusion drawn from sample measurements are usually expected to contain some errors – Type I error • To reject the null hypothesis when the null hypothesis is actually true! • Level of significance, specifies a limit on the probability of committing Type I error – Rejection region is different for a different value of 10 HYPOTHESIS TESTING SOME TERMS • Rejection Region – If the test statistic lies in the rejection region, the null hypothesis is rejected, else the null hypothesis is not rejected (not rejected accepted) – The rejection regions for z-test are shown below: z z / 2 • Two-tail test: reject the null hypothesis if • Right-tail test: reject the null hypothesis if z z • Left-tail test: reject the null hypothesis if z z – Where z is the test statistic, is the level of significance, and recall from Chapter 6 that zA is that value of z for which area on the right is A. 11 HYPOTHESIS TESTING SOME TERMS f(x) • Rejection Region Two -tail rejection region x 2 n 2 z / 2 z / 2 12 HYPOTHESIS TESTING SOME TERMS Left -tail rejection region x n f(x) f(x) • Rejection Region Right -tail rejection region x n z z 13 HYPOTHESIS TESTING SOME TERMS • Type I Error – Example: Suppose that a manufacturer of packaged cereals produces cereal boxes. Each box is expected to have a net weight of 100 gm. Periodically, samples are collected and the average weight of the sample is measured. It is possible that although the system is producing cereal boxes as usual, just because of some random variation, a sample may contain all boxes with weights less than 100 gm. Then, the manufacturer may be tempted to assume some problem with the system, stop the production and search for the problem. In this case, the sample data provides a false alarm and a Type 14 I error is committed. HYPOTHESIS TESTING SOME TERMS • Type II Error – Not to reject the null hypothesis when the null hypothesis is false! (the opposite of the Type I error). The probability of committing a Type II error is denoted by . – Example: consider the manufacturer of the packaged cereal again. Each cereal box is expected to have a net weight of 100 gm. But, due to some problems in the production system, the average weight is shifted to 98 gm. A Type II error is committed if a sample is collected with average weight nearly 100 gm. Notice that in such a case, the problem with the production system will not be detected by the sample! 15 TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN A z-test is used in the following context: • The measurements are normally distributed • The population standard deviation is known, • It is desirable to know if the population mean is – different from a given value (two-tail test) – less than a given value (left-tail test) – more than a given value (right-tail test) 16 TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN The test statistic and rejection region for the z-test are: • Test statistic: x z / n Where, x is the sample mean, is the population mean value stated in the null hypothesis, is the population standard deviation and n is the sample size. • Rejection region: – Two-tail test: reject the null hypothesis if z z / 2 – Right-tail test: reject the null hypothesis if z z – Left-tail test: reject the null hypothesis if z z where, is the level of significance. 17 TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN Example 4: A machine that produces ball bearings is set so that the average diameter is 0.60 inch. In a sample of 49 ball bearings, the mean diameter was found to be 0.61 inch. Assuming that the standard deviation is 0.035 can we conclude at the 5% significance level that the mean diameter is not 0.50 inch? HO : HA : Rejection region: Test statistic: Conclusion: f(x) x n 18 HYPOTHESIS TESTING INTERPRETATION • If the null hypothesis is rejected – Conclude that there is enough statistical evidence to infer that the alternative hypothesis is true • If the null hypothesis is not rejected – Conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true 19 TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN Example 5: A random sample of 100 observations from a normal population whose standard deviation is 50 produced a mean of 145. Does this statistic provide sufficient evidence at the 5% significance level to infer that population mean is more than 140? HO : Rejection region: Test statistic: Conclusion: x f(x) HA : n 20 TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN Example 6: A random sample of 100 observations from a normal population whose standard deviation is 50 produced a mean of 145. Does this statistic provide sufficient evidence at the 5% significance level to infer that population mean is less than 150? HO : Rejection region: Test statistic: Conclusion: x f(x) HA : n 21 READING AND EXERCISES • Sections 10.1-10.3: – Reading: pp. 327-343 – Exercises: 10.2,10.4,10.6 22