Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests

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1 Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests 1999 Prentice-Hall, Inc. Chap. 8-1

2 Chapter Topics Hypothesis Testing Methodology Z Test for the Mean (s 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 1999 Prentice-Hall, Inc. Chap. 8-2

3 What is a Hypothesis? 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! T/Maker Co Prentice-Hall, Inc. Chap. 8-3

4 The Null Hypothesis, H 0 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) Refers to the Status Quo Always contains the = sign The Null Hypothesis may or may not be rejected Prentice-Hall, Inc. Chap. 8-4

5 The Alternative Hypothesis, H 1 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 1999 Prentice-Hall, Inc. Chap. 8-5

6 Identify the Problem 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 Prentice-Hall, Inc. Chap. 8-6

7 Hypothesis Testing Process Assume the population mean age is 50. (Null Hypothesis) Is X 20 50? No, not likely! REJECT Null Hypothesis The Sample Mean Is 20 Population Sample 1999 Prentice-Hall, Inc. Chap. 8-7

8 Reason for Rejecting H 0 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 H 0 Sample Mean 1999 Prentice-Hall, Inc. Chap. 8-8

9 Level of Significance, a Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling Distribution Designated a (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 1999 Prentice-Hall, Inc. Chap. 8-9

10 Level of Significance, a and the Rejection Region H 0 : 3 H 1 : < 3 a Critical Value(s) H 0 : 3 H 1 : > 3 H 0 : 3 H 1 : 3 Rejection Regions a a/ Prentice-Hall, Inc. Chap. 8-10

11 Errors in Making Decisions Type I Error Reject True Null Hypothesis Has Serious Consequences Probability of Type I Error Is a Called Level of Significance Type II Error Do Not Reject False Null Hypothesis Probability of Type II Error Is b (Beta) 1999 Prentice-Hall, Inc. Chap. 8-11

12 Result Possibilities H 0 : Innocent Jury Trial Actual Situation Hypothesis Test Actual Situation Verdict Innocent Guilty Decision H 0 True H 0 False Innocent Correct Error Guilty Error Correct Do Not Reject H 0 Reject H a Type I Error ( a ) Type II Error ( b ) Power (1 - b ) 1999 Prentice-Hall, Inc. Chap. 8-12

13 a & b Have an Inverse Relationship Reduce probability of one error and the other one goes up. b a 1999 Prentice-Hall, Inc. Chap. 8-13

14 Factors Affecting Type II Error, b True Value of Population Parameter Increases When Difference Between Hypothesized Parameter & True Value Decreases Significance Level a Increases When a Decreases Population Standard Deviation s Increases When s Increases Sample Size n Increases When n Decreases 1999 Prentice-Hall, Inc. Chap b n b a b s

15 Z-Test Statistics (s Known) Convert Sample Statistic (e.g., X ) to Standardized Z Variable Z X s X X X s n Test Statistic Compare to Critical Z Value(s) If Z test Statistic falls in Critical Region, Reject H 0 ; Otherwise Do Not Reject H Prentice-Hall, Inc. Chap. 8-15

16 p Value Test 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 a Do Not Reject H 0 If p value < a, Reject H Prentice-Hall, Inc. Chap. 8-16

17 Hypothesis Testing: Steps Test the Assumption that the true mean # of TV sets in US homes is at least State H 0 H 0 : 3 2. State H 1 H 1 : < 3 3. Choose a a = Choose n n = Choose Test: Z Test (or p Value) 1999 Prentice-Hall, Inc. Chap. 8-17

18 Hypothesis Testing: Steps (continued) Test the Assumption that the average # of TV sets in US homes is at least Set Up Critical Value(s) Z = Collect Data 100 households surveyed 8. Compute Test Statistic Computed Test Stat.= Make Statistical Decision Reject Null Hypothesis 10. Express Decision The true mean # of TV set is less than 3 in the US households Prentice-Hall, Inc. Chap. 8-18

19 One-Tail Z Test for Mean (s Known) Assumptions Population Is Normally Distributed If Not Normal, use large samples Null Hypothesis Has or Sign Only Z Test Statistic: z x s x x x s n 1999 Prentice-Hall, Inc. Chap. 8-19

20 Rejection Region H 0 : H 1 : < 0 H 0 : 0 H 1 : > 0 Reject H 0 a Reject H 0 a 0 Must Be Significantly Below = 0 Z 0 Z Small values don t contradict H 0 Don t Reject H 0! 1999 Prentice-Hall, Inc. Chap. 8-20

21 Example: One Tail Test Does an average box of cereal contain more than 368 grams of cereal? A random sample _ of 25 boxes showed X = The company has specified s to be 15 grams. Test at the a0.05 level. 368 gm. H 0 : 368 H 1 : > Prentice-Hall, Inc. Chap. 8-21

22 Finding Critical Values: One Tail What Is Z Given a = 0.05? s Z = 1 a =.05 Standardized Normal Probability Table (Portion) Z Critical Value = Z Prentice-Hall, Inc. Chap. 8-22

23 Example Solution: One Tail H 0 : 368 H 1 : > 368 a = n = 25 Critical Value: Reject.05 Z Test Statistic: X Z 1.50 s n Decision: Do Not Reject at a =.05 Conclusion: No Evidence True Mean Is More than Prentice-Hall, Inc. Chap. 8-23

24 p Value Solution Use the alternative hypothesis to find the direction of the test. p Value is P(Z 1.50) = Prentice-Hall, Inc. Chap From Z Table: Lookup p Value.0668 Z Z Value of Sample Statistic

25 p Value Solution (p Value = ) (a = 0.05). Do Not Reject. p Value = Reject a = Z Test Statistic Is In the Do Not Reject Region 1999 Prentice-Hall, Inc. Chap. 8-25

26 Example: Two Tail Test Does an average box of cereal contains 368 grams of cereal? A random sample of 25 boxes showed X = The company has specified s to be 15 grams. Test at the a0.05 level. 368 gm. H 0 : 368 H 1 : Prentice-Hall, Inc. Chap. 8-26

27 Example Solution: Two Tail H 0 : 386 H 1 : 386 a = 0.05 n = 25 Critical Value: ± Reject.025 Z Test Statistic: X Z 1.50 s 15 n 25 Decision: Do Not Reject at a =.05 Conclusion: No Evidence that True Mean Is Not Prentice-Hall, Inc. Chap. 8-27

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

29 t-test: s Unknown Assumptions Population is normally distributed If not normal, only slightly skewed & a large sample taken Parametric test procedure t test statistic t X S n 1999 Prentice-Hall, Inc. Chap. 8-29

30 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 s 15. Test at the a0.01 level. s is not given, 368 gm. H 0 : 368 H 1 : > Prentice-Hall, Inc. Chap. 8-30

31 Example Solution: One Tail H 0 : 368 H 1 : > 368 a = 0.01 n = 36, df = 35 Critical Value: Reject Z t Test Statistic: Decision: Do Not Reject at a =.01 Conclusion: No Evidence that True Mean Is More than Prentice-Hall, Inc. Chap X S n

32 Proportions Involves categorical variables Fraction or % of population in a category If two categorical outcomes, binomial distribution Either possesses or doesn t possess the characteristic Sample proportion (p s ) p s X n number of successes sample size 1999 Prentice-Hall, Inc. Chap. 8-32

33 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 a =.05 significance level Prentice-Hall, Inc. Chap. 8-33

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

35 Chapter Summary Addressed Hypothesis Testing Methodology Performed Z Test for the Mean (s Known) Discussed p-value Approach to Hypothesis Testing Made Connection to Confidence Interval Estimation Performed One Tail and Two Tail Tests Performed t Test of Hypothesis for the Mean Performed Z Test of Hypothesis for the Proportion 1999 Prentice-Hall, Inc. Chap. 8-35