LESSON 20: HYPOTHESIS TESTING Outline Hypothesis testing Tests for the mean Tests for the proportion

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.

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.

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.

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 Note that since the null hypothesis always specifies a single value, none of the below may be a null hypothesis

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: One-tail (right-tail) test Two-tail test One-tail (left-tail) test Example 1 Example 2 Example 3

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.

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, 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, and the most appropriate test is a left-tail test.

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. For example, the test statistic for the z-test of mean is The test statistic is the same for both one-tail and two-tail 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)

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 it’s actually true! Type II error To accept the null hypothesis when it’s actually false! Level of significance,  specifies a limit on the probability of committing Type I error Rejection region is different for a different value of 

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: Two-tail test: reject the null hypothesis if Right-tail test: reject the null hypothesis if Left-tail test: reject the null hypothesis if Where z is the test statistic,  is the level of significance. Recall that zα is that value of z for which area on the right is α.

HYPOTHESIS TESTING SOME TERMS Rejection Region for two-tail z-test of mean

HYPOTHESIS TESTING SOME TERMS Rejection Region for one-tail z-test of mean

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 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!

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)

TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN The test statistic and rejection region for the z-test are: Test statistic: Where, is the sample mean,  is the population mean 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 Right-tail test: reject the null hypothesis if Left-tail test: reject the null hypothesis if where,  is the level of significance.

TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN For small population , and the test statistic for the z-test is: For single-decision procedure the rejection regions are stated in terms of z-statistic. However, for recurring-decision procedure, the rejection regions are stated in terms of the actual measurements, e.g.,

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. Assume that the standard deviation is 0.035. Does this statistic provide sufficient evidence at the 5% significance level to infer that the mean diameter is not 0.60 inch? Test statistic: Rejection region: Conclusion:

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

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? Test statistic: Rejection region: Conclusion:

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? Test statistic: Rejection region: Conclusion:

TESTING THE POPULATION MEAN WHEN THE POPULATION VARIANCE IS KNOWN Example 7: The following hypotheses are to be tested, with Assume σ = 25 and n = 100. The following decision rule applies: Compute Type I error probability when μ=110 and Type II error probability when μ=115.

INFERENCE ABOUT POPULATION MEAN WHEN THE POPULATION VARIANCE IS UNKNOWN If the population variance,  is not known, we cannot compute the z-statistic. However, we may compute a similar statistic, the t-statistic, that uses the sample standard deviation s in place of the population standard deviation : tα is that value of t for which the area to its right under the Student t-curve equals α. tα,df is that value of t for which the area to its right under the Student t-curve for degrees of freedom=df equals α. The value tα df is obtained from Table G, Appendix A, p. 541.

INFERENCE ABOUT POPULATION MEAN WHEN THE POPULATION VARIANCE IS UNKNOWN Test of Hypothesis about a population mean when the population variance is unknown: Test statistic: or Rejection region: Two-tail test: Right-tail test: Left-tail test:

INFERENCE ABOUT POPULATION MEAN WHEN THE POPULATION VARIANCE IS UNKNOWN Example 8: Let Suppose that will serve as test statistic. Assuming that a significance level of α=0.01 is desired, find the critical value for the sample mean and determine the decision rule. Should H0 be accepted or rejected if the computed sample mean turns out to be ?

INFERENCE ABOUT POPULATION MEAN WHEN THE POPULATION VARIANCE IS UNKNOWN Example 9: A random sample of 9 observations were drawn from a large population. These are: 11,9,5,7,1,2,10,6,3. Test to determine if we can infer at the 5% significance level that the population mean is less than 10. Test statistic: Rejection region: Conclusion: Answer:

INFERENCE ABOUT A POPULATION PROPORTION Let π = the population proportion p = the sample proportion n = the sample size Then, The expected number of successes = Standard deviation of successes = The expected sample proportion Standard deviation of the sample proportion

INFERENCE ABOUT A POPULATION PROPORTION Test of Hypothesis about a population proportion: Test statistic: Rejection region: Two-tail test: Right-tail test: Left-tail test:

INFERENCE ABOUT A POPULATION PROPORTION Example 10: Test the following hypothesis: Test statistic: Rejection region: Conclusion:

INFERENCE ABOUT A POPULATION PROPORTION Example 11: In a television commercial, the manufacturer of a toothpaste claims that more than seven out of 10 dentists recommend the ingredients in his product. To test the claim, a consumer protection group randomly samples 400 dentists and asks each one whether he or she recommend a toothpaste that contained the ingredients. A total of 290 dentists answered “Yes.” At the 5% significance level, can the consumer group infer that the claim is true?

Test statistic: Rejection region: Conclusion: Answer:

INFERENCE ABOUT A POPULATION PROPORTION Example 12: Test the following hypothesis: Test statistic: Rejection region: Conclusion:

READING AND EXERCISES Lesson 20 Reading: Section 11-1 to 11-3, pp. 344-389 Exercises: 11-4, 11-12, 11-14, 11-23, 11-31