Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Chapter 8 Hypothesis Testing 8-1 Review and Preview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim about a Proportion 8-4 Testing a Claim About a Mean 8-5 Testing a Claim About a Standard Deviation or Variance

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Definitions A hypothesis is a claim or statement about a property of a population. A hypothesis test is a procedure for testing a claim about a property of a population.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Null Hypothesis The null hypothesis (denoted by H 0 ) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value. We test the null hypothesis directly in the sense that we assume it is true and reach a conclusion to either reject H 0 or fail to reject H 0.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Alternative Hypothesis The alternative hypothesis (denoted by H 1 or H A ) is the statement that the parameter has a value that somehow differs from the null hypothesis. The symbolic form of the alternative hypothesis must use one of these symbols:, ≠.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Examples a) The mean annual income of employees who took a statistics course is greater than $60,000. b) The proportion of people aged 18 to 25 who currently use illicit drugs is equal to c) The standard deviation of human body temperatures is equal to 0.62 o F.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Examples d) The majority of college students have credit cards. e) The standard deviation of duration times (in seconds) of the Old Faithful geyser is less than 40 sec. f) The mean weight of airline passengers (including carry-on bags) is at most 195 lbs.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true. Test Statistic

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Test Statistic - Formulas Test statistic for proportion Test statistic for standard deviation Test statistic for mean

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Critical Region The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis. For example, see the red-shaded region in the previous figures.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Significance Level The significance level (denoted by α ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true (making the mistake of rejecting the null hypothesis when it is true). This is the same α introduced in Section 7-2. Common choices for α are 0.05, 0.01, and 0.10.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Find the Value of the Test Statistic, Then Find Either the P-Value or the Critical Value(s) First transform the relevant sample statistic to a standardized score called the test statistic. Then find the P-Value or the critical value(s).

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. P-Value The P-value (or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true. Critical region in the left tail: Critical region in the right tail: Critical region in two tails: P-value = area to the left of the test statistic P-value = area to the right of the test statistic P-value = twice the area in the tail beyond the test statistic

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Left-tailed Test  All α in the left tail

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Right-tailed Test  All α in the right tail

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Two-tailed Test  α is divided equally between the two tails of the critical region

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Decision Criterion: P-Value Method The null hypothesis is rejected if the P- value is very small. Using the significance level  : If P-value , then Reject H 0 If P-value > , then Fail to reject H 0

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Caution Don’t confuse a P-value with a proportion p. Know this distinction: P-value = probability of getting a test statistic at least as extreme as the one representing sample data p = population proportion

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example For each test, find the P-value and determine the decision about the null hypothesis if the test is conducted at a 0.05 level of significance. a) If H 1 :  < 95, and the test statistic has been found to be z = –1.89. b) If H 1 : p  0.6, and the test statistic has been found to be z = c) If H 1 :  > 41 and the test statistic has been found to be z = 2.05.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Restate the Decision Using Simple and Nontechnical Terms State a final conclusion that addresses the original claim with wording that can be understood by those without knowledge of statistical procedures.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Wording of Final Conclusion

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example: State the final conclusion. a) Original claim: The percentage of blue M&Ms is greater than 5%. Decision: Fail to reject the null hypothesis. b)Original claim: The percentage of Americans who know their credit score is equal to 20%. Decision: Fail to reject the null hypothesis. c) Original claim: The percentage of on-time U.S. airline flights is less than 75%. Decision: Reject the null hypothesis.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Type I Error A Type I error is the mistake of rejecting the null hypothesis when it is actually true. The symbol α is used to represent the probability of a type I error.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Type II Error A Type II error is the mistake of failing to reject the null hypothesis when it is actually false. The symbol β (beta) is used to represent the probability of a type II error.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Type I and Type II Errors

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Controlling Type I and Type II Errors For any fixed α, an increase in the sample size n will cause a decrease in β  For any fixed sample size n, a decrease in α will cause an increase in β. Conversely, an increase in α will cause a decrease in β. To decrease both α  and β, increase the sample size.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Hypothesis Test –P-Value Method State the null and alternative hypotheses in symbolic form. Select the level of significance  based on the seriousness of a type 1 error. The values of 0.05 and 0.01 are most common. Identify the statistic that is relevant to this test and determine its sampling distribution (such as normal z, t, or chi-square.) Find the test statistic and find the P-value. Draw a graph and show the test statistic and P-value. Technology may be used for this step. Reject H 0 if the P-value is less than or equal to the level of significance, , or Fail to reject H 0 if the P-value is greater than . Restate the previous decision in simple non-technical terms, and address the original claim.