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© 1999 Prentice-Hall, Inc. Chap. 8 - 1 Chapter Topics Hypothesis Testing Methodology Z Test for the Mean ( Known) p-Value Approach to Hypothesis Testing Connection to Confidence Interval Estimation One Tail Test t Test of Hypothesis for the Mean Z Test of Hypothesis for the Proportion

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© 1999 Prentice-Hall, Inc. Chap. 8 - 2 A hypothesis is an assumption about the population parameter. A parameter is a Population mean or proportion The parameter must be identified before analysis. I assume the mean GPA of this class is 3.5! © 1984-1994 T/Maker Co. What is a Hypothesis?

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© 1999 Prentice-Hall, Inc. Chap. 8 - 3 States the Assumption (numerical) to be tested e.g. The average # TV sets in US homes is at least 3 (H 0 : 3) Begin with the assumption that the null hypothesis is TRUE. (Similar to the notion of innocent until proven guilty) The Null Hypothesis, H 0 Refers to the Status Quo Always contains the ‘ = ‘ sign The Null Hypothesis may or may not be rejected.

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© 1999 Prentice-Hall, Inc. Chap. 8 - 4 Is the opposite of the null hypothesis e.g. The average # TV sets in US homes is less than 3 (H 1 : < 3) Challenges the Status Quo Never contains the ‘=‘ sign The Alternative Hypothesis may or may not be accepted The Alternative Hypothesis, H 1

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© 1999 Prentice-Hall, Inc. Chap. 8 - 5 Steps: State the Null Hypothesis (H 0 : 3) State its opposite, the Alternative Hypothesis (H 1 : < 3) Hypotheses are mutually exclusive & exhaustive Sometimes it is easier to form the alternative hypothesis first. Identify the Problem

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© 1999 Prentice-Hall, Inc. Chap. 8 - 6 Population Assume the population mean age is 50. (Null Hypothesis) REJECT The Sample Mean Is 20 Sample Null Hypothesis Hypothesis Testing Process No, not likely!

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© 1999 Prentice-Hall, Inc. Chap. 8 - 7 Sample Mean = 50 Sampling Distribution It is unlikely that we would get a sample mean of this value...... if in fact this were the population mean.... Therefore, we reject the null hypothesis that = 50. 20 H0H0 Reason for Rejecting H 0

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© 1999 Prentice-Hall, Inc. Chap. 8 - 8 Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling Distribution Designated (alpha) Typical values are 0.01, 0.05, 0.10 Selected by the Researcher at the Start Provides the Critical Value(s) of the Test Level of Significance,

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© 1999 Prentice-Hall, Inc. Chap. 8 - 9 Level of Significance, and the Rejection Region H 0 : 3 H 1 : < 3 0 0 0 H 0 : 3 H 1 : > 3 H 0 : 3 H 1 : 3 /2 Critical Value(s) Rejection Regions

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© 1999 Prentice-Hall, Inc. Chap. 8 - 10 Type I Error Reject True Null Hypothesis Has Serious Consequences Probability of Type I Error Is Called Level of Significance Type II Error Do Not Reject False Null Hypothesis Probability of Type II Error Is (Beta) Errors in Making Decisions

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© 1999 Prentice-Hall, Inc. Chap. 8 - 11 H 0 : Innocent Jury Trial Hypothesis Test Actual Situation Verdict InnocentGuilty Decision H 0 TrueH 0 False Innocent CorrectError Do Not Reject H 0 1 - Type II Error ( ) Guilty Error Correct Reject H 0 Type I Error ( ) Power (1 - ) Result Possibilities

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© 1999 Prentice-Hall, Inc. Chap. 8 - 12 Reduce probability of one error and the other one goes up. & Have an Inverse Relationship

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© 1999 Prentice-Hall, Inc. Chap. 8 - 13 Convert Sample Statistic (e.g., ) to Standardized Z Variable Compare to Critical Z Value(s) If Z test Statistic falls in Critical Region, Reject H 0 ; Otherwise Do Not Reject H 0 Z-Test Statistics ( Known) Test Statistic X

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© 1999 Prentice-Hall, Inc. Chap. 8 - 14 Probability of Obtaining a Test Statistic More Extreme or ) than Actual Sample Value Given H 0 Is True Called Observed Level of Significance Smallest Value of a H 0 Can Be Rejected Used to Make Rejection Decision If p value Do Not Reject H 0 If p value < , Reject H 0 p Value Test

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© 1999 Prentice-Hall, Inc. Chap. 8 - 15 1.State H 0 H 0 : 3 2.State H 1 H 1 : 3.Choose =.05 4.Choose n n = 100 5.Choose Test: Z Test (or p Value) Hypothesis Testing: Steps Test the Assumption that the true mean # of TV sets in US homes is at least 3.

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© 1999 Prentice-Hall, Inc. Chap. 8 - 16 6. Set Up Critical Value(s) Z = -1.645 7. Collect Data 100 households surveyed 8. Compute Test Statistic Computed Test Stat.= -2 9. Make Statistical Decision Reject Null Hypothesis 10. Express Decision The true mean # of TV set is less than 3 in the US households. Hypothesis Testing: Steps Test the Assumption that the average # of TV sets in US homes is at least 3. (continued)

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© 1999 Prentice-Hall, Inc. Chap. 8 - 17 Assumptions Population Is Normally Distributed If Not Normal, use large samples Null Hypothesis Has or Sign Only Z Test Statistic: One-Tail Z Test for Mean ( Known)

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© 1999 Prentice-Hall, Inc. Chap. 8 - 18 Z 0 Reject H 0 Z 0 0 H 0 : H 1 : < 0 H 0 : 0 H 1 : > 0 Must Be Significantly Below = 0 Small values don’t contradict H 0 Don’t Reject H 0 ! Rejection Region

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© 1999 Prentice-Hall, Inc. Chap. 8 - 19 Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has specified to be 15 grams. Test at the 0.05 level. 368 gm. Example: One Tail Test H 0 : 368 H 1 : > 368 _

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© 1999 Prentice-Hall, Inc. Chap. 8 - 20 Z.04.06 1.6.5495.5505.5515 1.7.5591.5599.5608 1.8.5671.5678.5686.5738.5750 Z0 Z = 1 1.645.50 -.05.45. 05 1.9.5744 Standardized Normal Probability Table (Portion) What Is Z Given = 0.05? =.05 Finding Critical Values: One Tail Critical Value = 1.645

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© 1999 Prentice-Hall, Inc. Chap. 8 - 21 = 0.025 n = 25 Critical Value: 1.645 Test Statistic: Decision: Conclusion: Do Not Reject at =.05 No Evidence True Mean Is More than 368 Z0 1.645.05 Reject Example Solution: One Tail H 0 : 368 H 1 : > 368

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© 1999 Prentice-Hall, Inc. Chap. 8 - 22 Z0 1.50 p Value. 0668 Z Value of Sample Statistic From Z Table: Lookup 1.50.9332 Use the alternative hypothesis to find the direction of the test. 1.0000 -.9332.0668 p Value is P(Z 1.50) = 0.0668 p Value Solution

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© 1999 Prentice-Hall, Inc. Chap. 8 - 23 0 1.50 Z Reject (p Value = 0.0668) ( = 0.05). Do Not Reject. p Value = 0.0668 = 0.05 Test Statistic Is In the Do Not Reject Region p Value Solution

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© 1999 Prentice-Hall, Inc. Chap. 8 - 24 Does an average box of cereal contains 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has specified to be 15 grams. Test at the 0.05 level. 368 gm. Example: Two Tail Test H 0 : 368 H 1 : 368

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© 1999 Prentice-Hall, Inc. Chap. 8 - 25 = 0.05 n = 25 Critical Value: ±1.96 Test Statistic: Decision: Conclusion: Do Not Reject at =.05 No Evidence that True Mean Is Not 368 Z 0 1.96.025 Reject Example Solution: Two Tail -1.96.025 H 0 : 386 H 1 : 386

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© 1999 Prentice-Hall, Inc. Chap. 8 - 26 Connection to Confidence Intervals For X = 372.5oz, = 15 and n = 25, The 95% Confidence Interval is: 372.5 - (1.96) 15/ 25 to 372.5 + (1.96) 15/ 25 or 366.62 378.38 If this interval contains the Hypothesized mean (368), we do not reject the null hypothesis. It does. Do not reject. _

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© 1999 Prentice-Hall, Inc. Chap. 8 - 27 Assumptions Population is normally distributed If not normal, only slightly skewed & a large sample taken Parametric test procedure t test statistic t-Test: Unknown

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© 1999 Prentice-Hall, Inc. Chap. 8 - 28 Example: One Tail t-Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and 15. Test at the 0.01 level. 368 gm. H 0 : 368 H 1 : 368 is not given,

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© 1999 Prentice-Hall, Inc. Chap. 8 - 29 Example:Z Test for Proportion Problem: A marketing company claims that it receives 4% responses from its Mailing. Approach: To test this claim, a random sample of 500 were surveyed with 25 responses. Solution: Test at the =.05 significance level.

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© 1999 Prentice-Hall, Inc. Chap. 8 - 30 =.05 n = 500 Do not reject at Do not reject at =.05 Z Test for Proportion: Solution H 0 : p .04 H 1 : p .04 Critical Values: 1.96 Test Statistic: Decision: Conclusion: We do not have sufficient evidence to reject the company’s claim of 4% response rate. Z p - p p (1 - p) n s =.04 -.05.04 (1 -.04) 500 = 1.14 Z0 Reject.025

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