Copyright © Cengage Learning. All rights reserved. 8 Introduction to Statistical Inferences

Copyright © Cengage Learning. All rights reserved. 8.3 The Nature of Hypothesis Testing

3 Hypothesis A statement that something is true. Statistical hypothesis test A process by which a decision is made between two opposing hypotheses. The two opposing hypotheses are formulated so that each hypothesis is the negation of the other. (That way, one of them is always true, and the other one is always false.) Then one hypothesis is tested in hopes that it can be shown to be a very improbable occurrence, thereby implying that the other hypothesis is likely the truth.

4 The Nature of Hypothesis Testing The two hypotheses involved in making a decision are known as the null hypothesis and the alternative hypothesis. Null hypothesis, H o The hypothesis we will test. Generally, this is a statement that a population parameter has a specific value. The null hypothesis is so named because it is the “starting point” for the investigation. (The phrase “there is no difference” is often used in its interpretation.)

5 The Nature of Hypothesis Testing Alternative hypothesis, H a A statement about the same population parameter that is used in the null hypothesis. Generally, this is a statement that specifies the population parameter has a value different, in some way, from the value given in the null hypothesis. The rejection of the null hypothesis will imply the likely truth of this alternative hypothesis.

6 Example 8 – Writing Hypotheses You are testing a new design for air bags used in automobiles, and you are concerned that they might not open properly. State the null and alternative hypotheses. Solution: The two opposing possibilities are “Bags open properly” and “Bags do not open properly.” Testing could produce evidence that discredits the hypothesis “Bags open properly”; plus your concern is that “Bags do not open properly.” Therefore, “Bags do not open properly” would become the alternative hypothesis and “Bags open properly” would be the null hypothesis.

7 The Nature of Hypothesis Testing The four possible outcomes that could result from the null hypothesis being either true or false and the decision being either to “reject H o ” or to “fail to reject H o ”. Table 8.3 shows these four possible outcomes. Four Possible Outcomes in a Hypothesis Test Table 8.3

8 The Nature of Hypothesis Testing A type A correct decision occurs when the null hypothesis is true and we decide in its favor. A type B correct decision occurs when the null hypothesis is false and the decision is in opposition to the null hypothesis. A type I error is committed when a true null hypothesis is rejected—that is, when the null hypothesis is true but we decide against it. A type II error is committed when we decide in favor of a null hypothesis that is actually false.

9 Example 12 – Describing the Possible Outcomes and Resulting Actions (On Hypothesis Tests) You suspect that a brand-name detergent outperforms the store’s brand of detergent, and you wish to test the two detergents because you would prefer to buy the cheaper store brand. Describe the four possible outcomes and the resulting actions that would occur for the hypothesis test. Solution: Your suspicion, “The brand-name detergent outperforms the store brand,” is the reason for the test and therefore becomes the alternative hypothesis.

10 Example 12 – Solution H o : “There is no difference in detergent performance.” H a : “The brand-name detergent performs better than the store brand.” cont’d

11 The Nature of Hypothesis Testing Notes 1.The truth of the situation is not known before the decision is made, the conclusion reached, and the resulting actions take place. The truth of H o may never be known. 2. The type II error often results in what represents a “lost opportunity”; lost in this situation is the chance to use a product that yields better results.

12 The Nature of Hypothesis Testing When a decision is made, it would be nice to always make the correct decision. This, however, is not possible in statistics because we make our decisions on the basis of sample information. The best we can hope for is to control the probability with which an error occurs.

13 The Nature of Hypothesis Testing The probability assigned to the type I error is  (called “alpha”). The probability of the type II error is  (called “beta”;  is the second letter of the Greek alphabet). See Table 8.4. Probability with which Decisions Occur Table 8.4

14 The Nature of Hypothesis Testing There is an interrelationship among the probability of the type I error (  ), the probability of the type II (  ), and the sample size (n). This is very much like the interrelationship among level of confidence, maximum error, and sample size.

15 The Nature of Hypothesis Testing Figure 8.8 shows the “three-way tug-of-war” among , , and n. If any one of the three is increased or decreased, it has an effect on one or both of the others. The “three-way tug-of-war” between ,  and n Figure 8.8

16 The Nature of Hypothesis Testing If any one of the three is increased or decreased, it has an effect on one or both of the others. The statistician’s job is thus to “balance” the three values of , , and n to achieve an acceptable testing situation. If  is reduced, then either  must increase or n must be increased; if  is decreased, then either  increases or n must be increased; if n is decreased, then either  increases or  increases. The choices for , , and n are definitely not arbitrary. At this time in our study of statistics, only the sample size, n, and , P(type I error), will be given and used to complete a hypothesis test.

17 The Nature of Hypothesis Testing , P(type II error), is further investigated in the section exercises but will not be utilized in this introduction to hypothesis testing. Level of significance  The probability of committing a type I error. Test statistic A random variable whose value is calculated from the sample data and is used in making the decision “reject H o ” or “fail to reject H o.” The value of the calculated test statistic is used in conjunction with a decision rule to determine either “reject H o ” or “fail to reject H o.”

18 The Nature of Hypothesis Testing This decision rule must be established prior to collecting the data; it specifies how you will reach the decision. The Conclusion a. If the decision is “reject H o,” then the conclusion should be worded something like, “There is sufficient evidence at the  level of significance to show that... [the meaning of the alternative hypothesis].” b. If the decision is “fail to reject H o,” then the conclusion should be worded something like, “There is not sufficient evidence at the a level of significance to show that... [the meaning of the alternative hypothesis].”

19 The Nature of Hypothesis Testing We must always remember that when the decision is made, nothing has been proved. Both decisions can lead to errors: “fail to reject H o ” could be a type II error (the lack of sufficient evidence has led to great parties being missed more than once), and “reject H o ” could be a type I error.