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ELEMENTARY STATISTICS Chapter 7 Hypothesis Testing MARIO F. TRIOLA EIGHTH Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION 1 Chapter 7 Hypothesis Testing 7-1 Overview 7-2 Fundamentals of Hypothesis Testing 7-3 Testing a Claim about a Mean: Large Samples 7-4 Testing a Claim about a Mean: Small Samples 7-5 Testing a Claim about a Proportion 7-6 Testing a Claim about a Standard Deviation Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2 7-1 Overview Definition Hypothesis in statistics, is a claim or statement about a property of a population Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3 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. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4 7-2 Fundamentals of Hypothesis Testing Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Figure 7-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Likely sample means µx = 98.6 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Figure 7-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Likely sample means µx = 98.6 z = - 1.96 or x = 98.48 z= 1.96 or x = 98.72 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7 Figure 7-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Sample data: z = - 6.64 or x = 98.20 Likely sample means µx = 98.6 z = - 1.96 or x = 98.48 z= 1.96 or x = 98.72 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8 Components of a Formal Hypothesis Test Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9 Null Hypothesis: H0 Statement about value of population parameter Must contain condition of equality =, , or Test the Null Hypothesis directly Reject H0 or fail to reject H0 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 Alternative Hypothesis: H1 Must be true if H0 is false , <, > ‘opposite’ of Null Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11 Note about Forming Your Own Claims (Hypotheses) If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12 Note about Testing the Validity of Someone Else’s Claim Someone else’s claim may become the null hypothesis (because it contains equality), and it sometimes becomes the alternative hypothesis (because it does not contain equality). Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13 Test Statistic a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14 Test Statistic a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis For large samples, testing claims about population means z= x - µx n Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15 Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16 Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17 Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18 Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Regions Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19 Significance Level denoted by the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. common choices are 0.05, 0.01, and 0.10 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20 Critical Value Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21 Critical Value Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Critical Value ( z score ) Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22 Critical Value Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Reject H0 Fail to reject H0 Critical Value ( z score ) Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23 Two-tailed,Right-tailed, Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 24 Two-tailed Test H0: µ = 100 H1: µ 100 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 25 Two-tailed Test H0: µ = 100 H1: µ 100 is divided equally between the two tails of the critical region Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 26 Two-tailed Test H0: µ = 100 H1: µ 100 is divided equally between the two tails of the critical region Means less than or greater than Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 27 Two-tailed Test H0: µ = 100 is divided equally between the two tails of the critical region H1: µ 100 Means less than or greater than Reject H0 Fail to reject H0 Reject H0 100 Values that differ significantly from 100 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 28 Right-tailed Test H0: µ 100 H1: µ > 100 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 29 Right-tailed Test H0: µ 100 H1: µ > 100 Points Right Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 30 Right-tailed Test H0: µ 100 H1: µ > 100 Points Right Fail to reject H0 100 Reject H0 Values that differ significantly from 100 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 31 Left-tailed Test H0: µ 100 H1: µ < 100 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 32 Left-tailed Test H0: µ 100 H1: µ < 100 Points Left Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 33 Left-tailed Test H0: µ 100 H1: µ < 100 Points Left Reject H0 Values that differ significantly from 100 Fail to reject H0 100 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 34 Conclusions in Hypothesis Testing always test the null hypothesis 1. Reject the H0 2. Fail to reject the H0 need to formulate correct wording of final conclusion See Figure 7-4 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 35 Wording of Final Conclusion FIGURE 7-4 Start Does the original claim contain the condition of equality Yes (Original claim contains equality and becomes H0) No Do you reject H0?. “There is sufficient evidence to warrant (Reject H0) rejection of the claim that. . . (original claim).” Yes No (Fail to reject H0) (Original claim does not contain equality and becomes H1) Do you reject H0? Yes (Reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” “The sample data supports the claim that . . . (original claim).” No (Fail to reject H0) (This is the only case in which the original claim is rejected). (This is the only case in which the original claim is supported). “There is not sufficient evidence to support the claim that. . . (original claim).” Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 36 Accept versus Fail to Reject some texts use “accept the null hypothesis we are not proving the null hypothesis sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect) Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 37 Type I Error The mistake of rejecting the null hypothesis when it is true. (alpha) is used to represent the probability of a type I error Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 38 Type II Error the mistake of failing to reject the null hypothesis when it is false. ß (beta) is used to represent the probability of a type II error Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 39 Table 7-2 Type I and Type II Errors True State of Nature We decide to reject the null hypothesis The null hypothesis is true The null hypothesis is false Type I error (rejecting a true null hypothesis) Correct decision Correct decision Type II error (rejecting a false null hypothesis) Decision We fail to reject the null hypothesis Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 40 Controlling Type I and Type II Errors For any fixed , an increase in the sample size n will cause a decrease in For any fixed sample size n , a decrease in will cause an increase in . Conversely, an increase in will cause a decrease in . To decrease both and , increase the sample size. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 41 Definition Power of a Hypothesis Test is the probability (1 - ) of rejecting a false null hypothesis, which is computed by using a particular significance level and a particular value of the mean that is an alternative to the value assumed true in the null hypothesis. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 42