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STATISTICS Chapter 8 Hypothesis Testing C.M. Pascual Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Fundamentals of Hypothesis Testing 8-3 Testing a Claim about a Mean: Large Samples 8-4 Testing a Claim about a Mean: Small Samples 8-5 Testing a Claim about a Proportion 8-6 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 8-1 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 8-2 Fundamentals of Hypothesis Testing Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Figure 8-1 Central Limit Theorem Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Figure 8-1 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 7 Figure 8-1 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 8 Figure 8-1 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 9 Components of a Formal Hypothesis Test Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 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 11 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 12 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 13 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 14 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 15 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 16 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 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 Region Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19 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 20 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 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 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 Critical Value ( z score ) Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23 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 24 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 25 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 26 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 27 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 28 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 29 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 30 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 31 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 32 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 33 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 34 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 35 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 8-4 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 36 Wording of Final Conclusion FIGURE 8-4 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 37 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 38 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 39 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 40 Table 8-2 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 41 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 42 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 43 Steps in Hypothesis Testing 1. State the null and alternative hypothesis; 2. Select the level of significance; 3. Determine the critical value and the rejection region/s; 4. State the decision rule; 5. Compute the test statistics; and 6. Make a decision, whether to reject or not to reject the null hypothesis. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 44 Example 1 • A manufacturer claims that the average lifetime of his lightbulbs is 3 years or 36 months. The stabdard deviation is 8 months. Fifty (50) bulbs are selected, and the average lifetime is found to be 32 months. Should the manufacturer’s statement be rejected at = 001? Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 45 Example 1 • Solution: • Step 1. State the hypothesis: – Ho: µ = 36 months – Ha : µ 36 months • Step 2. Level of significance =001 • Step 3. Determine critical values and rejection region Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 46 Example 1 • Solution: • Step 3. Determine critical values and rejection region – Z = +/- 2.575 (from Appendix B of z values) • Step 4. State the decision rule – Reject the null hypothesis if Zc > 2.575 or Zc = 2.575 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 47 Example 1 • Solution: • Step 5. Compute the test statistic. zc = x - µx n Zc = (32-36)/ (8/(50)0.5 = - 3.54 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 48 Example 1 • Solution: • Step 6. Make a decision. Zc = - 3.54 is less than Z = -2.575 And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 49 Example 1 • Solution: • Step 6. Make a decision. Zc = - 3.54 is less than Z = -2.575 And it falls in the rejection region in the left tail. Therefore, reject Ho and conclude that the average lifetime of lightbulbs is not equal to 36 months. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 50 Example 2 • A test on car braking reaction times for men between 18 to 36 years old have produced a mean and standard deviation of 0.610 second and 0.123 second, respectively. When 40 male drivers of this age group were randomly selected and tested for their breaking reaction times, a mean of 0.587 second came out. At the = 0.10, test the claim of the driving instructor that his graduates had faster reaction times. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 51 Example 2 • Solution: • Step 1. State the hypothesis: – Ho: µ = 0.610 second – Ha: µ < 0.610 second • Step 2. Level of significance =010 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 52 Example 2 • Solution: • Step 3. Determine critical values and rejection region Z = - 1/.28 (from Appendix B of z values) • Step 4. State the decision rule – Reject the null hypothesis if Zc < - 1.28 . Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 53 Example 2 • Solution: • Step 5. Compute the test statistic. zc = x - µx n Zc = (0.587-0.610)/ (0.123/(40)0.5 = - 1.18 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 54 Example 2 • Solution: • Step 6. – Since the test statistics falls within the non-critical region, do not reject Ho. There is enough evidence to support the instructor’s claim.; accept Ho. . Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 55 Test on Small Sample Mean • The t-test is a statistical test for the mean of a population and is used when the population is normally distributed, σ is unknown, and n < 30. The formula for the t-test with degrees of freedom are d.f. = n – 1 is t= x-µ s n Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 56 Example 3 • In order to increase customer service, a muffler repair shop claim its mechanics can replace a muffler in 12 minutes. A time management specialist selected 6 repair jobs and found their mean time to be 11.6 minutes. The standard deviation of the sample was 2.1 minutes. A = 0.025, is there enough evidence to conclude that the mean time in changing a muffler is less than 12 minutes? Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 57 Example 3 Solution 1. State the hypothesis: 1. Ho: µ = 12 2. Ha: µ < 12 2. Step 2. Level of significance =0025 3. Step 3. Since and d.f. = 6 – 1 = 5, then at = 0025,Appendix C at t-value = - 2.571 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 58 Example 1 Solution 4. Step 4. Reject Ho if tc < - 2.571 5. Compute for the test statistic t= x-µ s Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 59 Example 3 t = (11.6 – 12)/(2.1(6)0.5 = - 0.47 • Step 6. Since the critical value fall within the non-critical region, do not reject Ho. Accept Ho. Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 60 Sit Work (Submit after class) 1. A diet clinic states that there is an average loss of 24 pounds for those who stay on the program for 20 weeks. The standard deviation is 5 pounds. The clinic tries a new diet, reducing salt intake to see whether that strategy will produce a greater weight loss. A group of 40 volunteers loses an average of 16.3 pounds each over 20 weeks. Should the clinic change the new diet? Use = 0.05 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 61 Assignment (Submit next meeting) 2. A recent survey stated that household received an average 37 telephone calls per month. To test the claim, a researcher surveyed 29 households and found that the average number of calls was 34.9. The standard deviation of the sample was 6. At = 0.05, can the claim be substantiated? Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 62