Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests 1999 Prentice-Hall, Inc. Chap. 8-1
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
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! 1984-1994 T/Maker Co. 1999 Prentice-Hall, Inc. Chap. 8-3
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. 1999 Prentice-Hall, Inc. Chap. 8-4
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
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. 1999 Prentice-Hall, Inc. Chap. 8-6
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
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. 20 = 50 H 0 Sample Mean 1999 Prentice-Hall, Inc. Chap. 8-8
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
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 0 0 0 a a/2 1999 Prentice-Hall, Inc. Chap. 8-10
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
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 0 1 - a Type I Error ( a ) Type II Error ( b ) Power (1 - b ) 1999 Prentice-Hall, Inc. Chap. 8-12
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
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. 8-14 b n b a b s
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 0 1999 Prentice-Hall, Inc. Chap. 8-15
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 0 1999 Prentice-Hall, Inc. Chap. 8-16
Hypothesis Testing: Steps Test the Assumption that the true mean # of TV sets in US homes is at least 3. 1. State H 0 H 0 : 3 2. State H 1 H 1 : < 3 3. Choose a a =.05 4. Choose n n = 100 5. Choose Test: Z Test (or p Value) 1999 Prentice-Hall, Inc. Chap. 8-17
Hypothesis Testing: Steps (continued) Test the Assumption that the average # of TV sets in US homes is at least 3. 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. 1999 Prentice-Hall, Inc. Chap. 8-18
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
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
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 = 372.5. The company has specified s to be 15 grams. Test at the a0.05 level. 368 gm. H 0 : 368 H 1 : > 368 1999 Prentice-Hall, Inc. Chap. 8-21
Finding Critical Values: One Tail What Is Z Given a = 0.05?.50 -.05.45 s Z = 1 a =.05 Standardized Normal Probability Table (Portion) Z.04.05.06 1.6.5495.5505.5515 1.7.5591.5599.5608 0 Critical Value = 1.645 1.645 Z 1.8.5671.5678.5686 1.9.5738.5744.5750 1999 Prentice-Hall, Inc. Chap. 8-22
Example Solution: One Tail H 0 : 368 H 1 : > 368 a = 0.025 n = 25 Critical Value: 1.645 0 1.645 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 368 1999 Prentice-Hall, Inc. Chap. 8-23
p Value Solution Use the alternative hypothesis to find the direction of the test. p Value is P(Z 1.50) = 0.0668.9332 1999 Prentice-Hall, Inc. Chap. 8-24 0 From Z Table: Lookup 1.50 1.50 p Value.0668 Z 1.0000 -.9332.0668 Z Value of Sample Statistic
p Value Solution (p Value = 0.0668) (a = 0.05). Do Not Reject. p Value = 0.0668 Reject a = 0.05 0 1.50 Z Test Statistic Is In the Do Not Reject Region 1999 Prentice-Hall, Inc. Chap. 8-25
Example: Two Tail Test 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 s to be 15 grams. Test at the a0.05 level. 368 gm. H 0 : 368 H 1 : 368 1999 Prentice-Hall, Inc. Chap. 8-26
Example Solution: Two Tail H 0 : 386 H 1 : 386 a = 0.05 n = 25 Critical Value: ±1.96.025-1.96 0 1.96 Reject.025 Z Test Statistic: X 372.5 368 Z 1.50 s 15 n 25 Decision: Do Not Reject at a =.05 Conclusion: No Evidence that True Mean Is Not 368 1999 Prentice-Hall, Inc. Chap. 8-27
Connection to Confidence Intervals _ For X = 372.5oz, s = 15 and n = 25, The 95% Confidence Interval is: 372.5 - (1.96) 15/ 25 to 372.5 + (1.96) 15/ 25 1999 Prentice-Hall, Inc. Chap. 8-28 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.
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
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 : > 368 1999 Prentice-Hall, Inc. Chap. 8-30
Example Solution: One Tail H 0 : 368 H 1 : > 368 a = 0.01 n = 36, df = 35 Critical Value: 2.4377 Reject.01 0 2.4377 Z t Test Statistic: 372.5 15 Decision: Do Not Reject at a =.01 Conclusion: No Evidence that True Mean Is More than 368 1999 Prentice-Hall, Inc. Chap. 8-31 X S n 36 368 1.80
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
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. 1999 Prentice-Hall, Inc. Chap. 8-33
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. 1999 Prentice-Hall, Inc. Chap. 8-34 s =.04 -.05.04 (1 -.04) 500 = 1.14
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