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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-1 Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics for Managers Using Microsoft ® Excel 4 th Edition

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-2 Chapter Goals After completing this chapter, you should be able to: Formulate null and alternative hypotheses for applications involving a single population mean or proportion Formulate a decision rule for testing a hypothesis Know how to use the p-value approaches to test the null hypothesis for both mean and proportion problems Know what Type I and Type II errors are

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-3 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean population proportion Example: The mean monthly cell phone bill of this city is μ = $42 Example: The proportion of adults in this city with cell phones is p =.68

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-4 States the assumption to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) Is always about a population parameter, not about a sample statistic The Null Hypothesis, H 0

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-5 The Null Hypothesis, H 0 Begins with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Refers to the status quo Always contains “=”, “≤” or “ ” sign May or may not be rejected (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-6 The Alternative Hypothesis, H 1 Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S. homes is not equal to 3 ( H 1 : μ ≠ 3 ) Challenges the status quo Never contains the “=”, “≤” or “ ” sign Is generally the hypothesis that is believed (or needs to be supported) by the researcher

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-7 Hypothesis Testing We assume the null hypothesis is true If the null hypothesis is rejected we have proven the alternate hypothesis If the null hypothesis is not rejected we have proven nothing as the sample size may have been to small

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Population Claim: the population mean age is 50. (Null Hypothesis: REJECT Suppose the sample mean age is 20: X = 20 Sample Null Hypothesis 20 likely if μ = 50? Is Hypothesis Testing Process If not likely, Now select a random sample H 0 : μ = 50 ) X

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-9 Do not reject H 0 Reject H 0 There are two cutoff values (critical values), defining the regions of rejection Sampling Distribution of /2 0 H 0 : μ = 50 H 1 : μ 50 /2 Lower critical value Upper critical value 50 X 20 Likely Sample Results

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-10 Level of Significance, Defines the unlikely values of the sample statistic if the null hypothesis is true Defines rejection region of the sampling distribution Is designated by , (level of significance) Typical values are.01,.05, or.10 Is the compliment of the confidence coefficient Is selected by the researcher before sampling Provides the critical value of the test

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-11 Level of Significance and the Rejection Region H 0 : μ ≥ 3 H 1 : μ < 3 0 H 0 : μ ≤ 3 H 1 : μ > 3 Represents critical value Lower tail test Level of significance = 0 Upper tail test Two tailed test Rejection region is shaded /2 0 H 0 : μ = 3 H 1 : μ ≠ 3

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-12 Type I Error When a true null hypothesis is rejected The probability of a Type I Error is Called level of significance of the test Set by researcher in advance Type II Error Failure to reject a false null hypothesis The probability of a Type II Error is β Errors in Making Decisions

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-13 Example The Truth Verdict InnocentNo error Type II Error GuiltyType I Error Possible Jury Trial Outcomes Guilty Innocent No Error

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-14 Outcomes and Probabilities Actual Situation Decision Do Not Reject H 0 No error (1 - ) Type II Error ( β ) Reject H 0 Type I Error ( ) Possible Hypothesis Test Outcomes H 0 False H 0 True Key: Outcome (Probability) No Error ( 1 - β )

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-15 Type I & II Error Relationship Type I and Type II errors can not happen at the same time Type I error can only occur if H 0 is true Type II error can only occur if H 0 is false If Type I error probability ( ), then Type II error probability ( β )

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-16 p-Value Approach to Testing p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H 0 is true Also called observed level of significance

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-17 p-Value Approach to Testing Convert Sample Statistic (e.g. ) to Test Statistic (e.g. t statistic ) Obtain the p-value from a table or computer Compare the p-value with If p-value < , reject H 0 If p-value , do not reject H 0 X (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-18 9 Steps in Hypothesis Testing 1. State the null hypothesis, H 0 2.State the alternative hypotheses, H 1 3. Choose the level of significance, α 4. Choose the sample size, n 5. Determine the appropriate test statistic to use 6. Collect the data 7Compute the p-value for the test statistic from the sample result 8. Make the statistical decision: Reject H 0 if the p-value is less than alpha 9.Express the conclusion in the context of the problem

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-19 Hypothesis Tests for the Mean Known Unknown Hypothesis Tests for

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-20 Hypothesis Testing Example Test the claim that the true mean # of TV sets in U.S. homes is equal to 3. 1-2. State the appropriate null and alternative hypotheses H 0 : μ = 3 H 1 : μ ≠ 3 (This is a two tailed test) 3. Specify the desired level of significance Suppose that =.05 is chosen for this test 4. Choose a sample size Suppose a sample of size n = 100 is selected

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-21 5. Determine the appropriate Test σ is unknown so this is a t test 6. Collect the data Suppose the sample results are n = 100, = 2.84 s = 0.8 7. So the test statistic is: The p value for n=100, =.05, t=-2 is.048 Hypothesis Testing Example (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-22 Reject H 0 Do not reject H 0 8. Is the test statistic in the rejection region? =.05/2 -t= -1.98 0 Reject H 0 if p is < alpha; otherwise do not reject H 0 Hypothesis Testing Example (continued) =.05/2 Reject H 0 +t= +1.98 Here, t = -2.0 < -1.98, so the test statistic is in the rejection region The p-value.048 is < alpha.05, we reject the null hypothesis

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-23 9. Express the conclusion in the context of the problem Since The p-value.048 is < alpha.05, we have rejected the null hypothesis Thereby proving the alternate hypothesis Conclusion: There is sufficient evidence that the mean number of TVs in U.S. homes is not equal to 3 Hypothesis Testing Example (continued) If we had failed to reject the null hypothesis the conclusion would have been: There is not sufficient evidence to reject the claim that the mean number of TVs in U.S. home is 3

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-24 One Tail Tests In many cases, the alternative hypothesis focuses on a particular direction H 0 : μ ≥ 3 H 1 : μ < 3 H 0 : μ ≤ 3 H 1 : μ > 3 This is a lower tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 This is an upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-25 Reject H 0 Do not reject H 0 There is only one critical value, since the rejection area is in only one tail Lower Tail Tests -t 3 H 0 : μ ≥ 3 H 1 : μ < 3 Critical value

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-26 Reject H 0 Do not reject H 0 Upper Tail Tests tαtα 3 H 0 : μ ≤ 3 H 1 : μ > 3 There is only one critical value, since the rejection area is in only one tail Critical value t

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-27 Assumptions of the One-Sample t Test The data is randomly selected The population is normally distributed or the sample size is over 30 and the population is not highly skewed

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-28 Hypothesis Tests for Proportions Involves categorical values Two possible outcomes “Success” (possesses a certain characteristic) “Failure” (does not possesses that characteristic) Fraction or proportion of the population in the “success” category is denoted by p

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-29 Proportions Sample proportion in the success category is denoted by p s When both np and n(1-p) are at least 5, p s can be approximated by a normal distribution with mean and standard deviation (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-30 The sampling distribution of p s is approximately normal, so the test statistic is a Z value: Hypothesis Tests for Proportions np 5 and n(1-p) 5 Hypothesis Tests for p np < 5 or n(1-p) < 5 Not discussed in this chapter

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-31 An equivalent form to the last slide, but in terms of the number of successes, X: Z Test for Proportion in Terms of Number of Successes X 5 and n-X 5 Hypothesis Tests for X X < 5 or n-X < 5 Not discussed in this chapter

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-32 Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the =.05 significance level. Check: n p = (500)(.08) = 40 n(1-p) = (500)(.92) = 460

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-33 Z Test for Proportion: Solution =.05 n = 500, p s =.05 p-value for -2.27 is.0134 Decision: Reject H 0 at =.05 H 0 : p =.08 H 1 : p .08 Critical Values: ± 1.96 Test Statistic: Conclusion: z 0 Reject.025 1.96 -2.47 There is sufficient evidence to reject the company’s claim of 8% response rate. -1.96

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-34 Using PHStat Options

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-35 Sample PHStat Output Input Output

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-36 Potential Pitfalls and Ethical Considerations Use randomly collected data to reduce selection biases Do not use human subjects without informed consent Choose the level of significance, α, before data collection Do not employ “data snooping” to choose between one- tail and two-tail test, or to determine the level of significance Do not practice “data cleansing” to hide observations that do not support a stated hypothesis Report all pertinent findings

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-37 Chapter Summary Addressed hypothesis testing methodology Discussed critical value and p–value approaches to hypothesis testing Discussed type 1 and Type2 errors Performed two tailed t test for the mean (σ unknown) Performed Z test for the proportion Discussed one-tail and two-tail tests Addressed pitfalls and ethical issues

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-38 Answer Sheet for All Problems ___________ Null Hypothesis ___________ Alternate Hypothesis ___________ Alpha ___________ p-value ___________ Decision (reject or do not reject) Conclusion:

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