HOMEWORK (due Fri, Nov 19): Chapter 12: #62, 83, 101

Transcription

1 Today: Section 2.2, Lesson 3: What can go wrong with hyothesis testing Section 2.4: Hyothesis tests for difference in two roortions ANNOUNCEMENTS: No discussion today. Check your grades on eee and notify me if any of them are incorrect. Quiz #7 begins Wed after class and ends Friday. Quiz #8 begins Wed before Thanksgiving and ends on Monday after Thanksgiving. That is the last quiz. Jason Kramer will give the lecture this Friday. HOMEWORK (due Fri, Nov 9): Chater 2: #62, 83, 0

2 REVIEW OF HYPOTHESIS TEST FOR ONE PROPORTION ONE-SIDED TEST, WITH PICTURE H 0 : = 0 versus Ha: > 0 Reject H 0 if -value <.05. For what values of z does that haen? Notation: level of significance = α (alha, usually.05) -value <.05 corresonds with z >.645 Normal, Mean=0, StDev= Do not reject null hyothesis 0 Value of test statistic Reject null hyothesis

3 AN ILLUSTRATION OF WHAT HAPPENS WHEN Ha IS TRUE A Secific Examle Suose the truth for the oulation roortion is one standard deviation above the null value 0. Then the mean for the standardized scores will be instead of 0. How often would we (correctly) reject the null hyothesis in that case? Answer (urle region) is Picture when the true oulation roortion is s.d. above null value Normal, Mean=, StDev= Do not reject null hyothesis Value of test statistic Reject null hyothesis

4 Section 2.2, Lesson 3 What Can Go Wrong in Hyothesis Testing: The Two Tyes of Errors and Their Probabilities Examle: Case Study.6, asirin and heart attacks. Found statistically significant relationshi; -value was < Possible errors: Tye error (false ositive) occurs when: Null hyothesis is actually true, but Conclusion of test is to Reject H 0 and accet H a Tye 2 error (false negative) occurs when: Alternative hyothesis is actually true, but Conclusion is that we cannot reject H 0

5 Heart attack and asirin examle: Null hyothesis: The roortion of men who would have heart attacks if taking asirin = the roortion of men who would have heart attacks if taking lacebo. Alternative hyothesis: The heart attack roortion is lower if men were to take asirin than if they were not to take asirin. Tye error (false ositive): Occurs if there really is no relationshi between taking asirin and heart attack revention, but we conclude that there is a relationshi. Consequence: Good for asirin comanies! Possible bad side effects from asirin, with no redeeming value.

6 Tye 2 error (false negative): Occurs if there really is a relationshi but study failed to find it. Consequence: Miss out on recommending something that could save lives! Which tye of error is more serious? Probably all agree that Tye 2 is more serious. Which could have occurred? Tye error could have occurred. Tye 2 could not have occurred, because we did find a significant relationshi.

7 Asirin Examle: Consequences of the decisions Decision: Truth: H 0 : Asirin doesn t work Ha: Asirin works Which error can occur? Don t conclude asirin works OK Tye 2 error: Asirin could save lives but we don t recognize benefits Tye 2 can only occur if H 0 is not Reject H 0, Conclude asirin works Tye error: Peole take asirin needlessly; may suffer side effects OK Tye can only occur if H 0 is rejected. Which error can occur: Tye error can only occur if asirin doesn t work. Tye 2 error can only occur if asirin does work. rejected. Note that because H 0 was rejected in this study, we could only have made a Tye error, not a Tye 2 error.

8 Some analogies to hyothesis testing: Analogy : Courtroom: Null hyothesis: Defendant is innocent. Alternative hyothesis: Defendant is guilty Note that the two ossible conclusions are not guilty and guilty. The conclusion not guilty is equivalent to don t reject H 0. We don t say defendant is innocent just like we don t accet H 0 in hyothesis testing. Tye error is when defendant is innocent but gets convicted Tye 2 error is when defendant is guilty but does not get convicted. Which one is more serious??

9 Analogy 2: Medical test Null hyothesis: You do not have the disease Alternative hyothesis: You have the disease Tye error: You don't have disease, but test says you do; a "false ositive" Tye 2 error: You do have disease, but test says you do not; a "false negative" Which is more serious??

10 Notes and Definitions: Probability related to Tye error: The conditional robability of making a Tye error, given that H 0 is true, is the level of significance α. In most cases, this is.05. However, it should be adjusted to be lower (.0 is common) if a Tye error is more serious than a Tye 2 error. In robability notation: P(Reject H 0 H 0 is true) = α, usually.05. Normal, Mean=0, StDev= Do not reject null hyothesis 0 Value of test statistic Reject null hyothesis

11 Probability related to Tye 2 error and Power: The conditional robability of making a correct decision, given that H a is true is called the ower of the test. This can only be calculated if a secific value in Ha is secified. Conditional robability of making a Tye 2 error = ower. P(Reject H 0 Ha is true) = ower. P(Do not reject H 0 Ha is true) = ower = β =P(Tye 2 error). Picture when the true oulation roortion is s.d. above null value Normal, Mean=, StDev= Probability of tye 2 error = =.7405 Do not reject null hyothesis = ower.645 Value of test statistic Reject null hyothesis

12 How can we increase ower and decrease P(Tye 2 error)? Power increases if: Samle size is increased (because having more evidence makes it easier to show that the alternative hyothesis is true, if it really is) The level of significance α is increased (because it s easier to reject H 0 when the cutoff oint for the -value is larger) The actual difference between the samle estimate and the null value increases (because it s easier to detect a true difference if it s large) We have no control over this one! Trade-off must be taken into account when choosing α. If α is small it s harder to reject H 0. If α is large it s easier to reject H 0 : If Tye error is more serious, use smaller α. If Tye 2 error is more serious, use larger α.

13 Ways to icture the errors: Truth, decisions, consequences, conditional (row) robabilities: Decision: Truth: Don t reject H 0 Reject H 0 Error can occur: Correct Tye error Tye error can only H 0 α α Tye 2 error Correct H a β = ower ower Error can occur: H 0 not rejected H 0 rejected occur if H 0 true Tye 2 error can only occur if H a is true.

14 SECTION 2.4: Test for difference in 2 roortions Reminder from when we started Chater 9 Five situations we will cover for the rest of this quarter: Parameter name and descrition Poulation arameter Samle statistic For Categorical Variables: One oulation roortion (or robability) ˆ Difference in two oulation roortions 2 ˆ ˆ 2 For Quantitative Variables: One oulation mean µ x Poulation mean of aired differences (deendent samles, aired) µ d d Difference in two oulation means µ (indeendent samles) µ 2 x x2 For each situation will we: Learn about the samling distribution for the samle statistic Learn how to find a confidence interval for the true value of the arameter Test hyotheses about the true value of the arameter

15 Comaring two roortions from indeendent samles Reminder on how we get indeendent samles (Lecture 9): Random samles taken searately from two oulations and same resonse variable is recorded. Examle: Comare roortions who think global warming is a roblem, in two different years. One random samle taken and a variable recorded, but units are categorized to form two oulations. Examle: Comare 2 and over with under 2 for roortion who drink alcohol. Particiants randomly assigned to one of two treatment conditions, and same resonse variable is recorded. Examle: Comare asirin and lacebo grous for roortions who had heart attacks.

16 Hyothesis Test for Difference in Two Proortions Examle: (Source: htt:// Poll taken in June 2006, just after the May release of An Inconvenient Truth Asked 500 eole In your view, is global warming a very serious roblem, somewhat serious, not too serious, or not a roblem? Results: 65/500 =.4 or 4% answered Very serious Poll taken again with different 500 eole in October, Results: 525/500 =.35 or 35% answered Very serious. Question: Did the oulation roortion that thinks it s very serious go down from 2006 to 2009, or is it chance fluctuation?

17 Notation and numbers for the Examle: Poulation arameter of interest is 2 where: = 2 = roortion of all US adults in May 2006 who thought global warming was a serious roblem. roortion of all US adults in Oct 2009 who thought global warming was a serious roblem. ˆ = samle estimate from May 2006 = X /n = 65/500 =.4 ˆ 2 = samle estimate from Oct 2009 = X 2 /n 2 = 525/500 =.35 Samle statistic is ˆ ˆ 2 =.4.35 =. 06

18 Five stes to hyothesis testing for difference in 2 roortions: See Summary Box on ages STEP : Determine the null and alternative hyotheses. Null hyothesis is H 0 : 2 = 0 (or = 2 ); null value = 0 Alternative hyothesis is one of these, based on context: H a : 2 0 (or 2 ) H a : 2 > 0 (or > 2 ) H a : 2 < 0 (or < 2 ) EXAMPLE: Did the oulation roortion who think global warming is very serious dro from 2006 to 2009? This is the alternative hyothesis. (Note that it s a one-sided test.) H 0 : 2 = 0 (no actual change in oulation roortions) H a : 2 > 0 (or > 2 ; 2006 roortion > 2009 roortion)

19 STEP 2: Verify data conditions. If met, summarize data into test statistic. For Difference in Two Proortions: Data conditions: nˆ and n( ˆ ) are both at least 0 for both samles. Test statistic: ˆ z = samle statistic null value (null) standard error Samle statistic = 2 Null value = 0 Null standard error: Comuted assuming null hyothesis is true. If null hyothesis is true, then = 2 We get an estimate for the common value of using both samles, then use that in the standard error formula. Details on next age. ˆ

20 combined samle sizes combined successes ˆ 2 2 = + + = n n X X Null standard error = estimate of ) ( ) ( n n +, using combined estimate ˆ in lace of both and 2. So the test statistic is: ) ˆ)( ˆ( 0 ) ˆ ˆ ( ˆ) ˆ( ˆ) ˆ( 0 ) ˆ ˆ ( n n n n z + = + =

21 Ste 2 for the Examle: Data conditions are met, since both samle sizes are 500. X + X combined successes = = = = = n + n combined samle sizes ˆ.38 2 Null standard error = (. 38)(.38)( + ) = Test statistic: z = samle statistic null value (null) standard error = = 3.39

22 Pictures: On left: Samling distribution of ˆ ˆ 2 when oulation roortions are equal and samle sizes are both 500, showing where the observed value of.06 falls. On right: The same icture, converted to z-scores. Samling distribution for hat - 2hat, when =2, n=n2=500 Normal, Mean=0, StDev=0.077 Standard normal distribution Normal, Mean=0, StDev= Possible values of hat - 2hat Values of z 3.39 Note that area above ˆ ˆ 2 = 0.06 is so small you can t see it!

23 STEP 3: Assuming the null hyothesis is true, find the -value. General: -value = the robability of a test statistic as extreme as the one observed or more so, in the direction of H a, if the null hyothesis is true. Difference in two roortions, same idea as one roortion. Deends on the alternative hyothesis. See ictures on. 57 Alternative hyothesis: -value is: H a : 2 > 0 (a one-sided hyothesis) Area above the test statistic z H a : 2 < 0 (a one-sided hyothesis) H a : 2 0 (a two-sided hyothesis) Area below the test statistic z 2 the area above z = area in tails beyond z and z

24 Examle: Alternative hyothesis is one-sided H a : 2 > 0 -value = Area above the test statistic z = 3.39 From Table A., -value = area above 3.39 =.9997 = STEP 4: Decide whether or not the result is statistically significant based on the -value. Examle: Use α of.05, as usual -value =.0003 <.05, so: Reject the null hyothesis. Accet the alternative hyothesis The result is statistically significant

25 Ste 5: Reort the conclusion in the context of the situation. Examle: Conclusion: From May 2006 to October 2009 there was a statistically significant decrease in the roortion of US adults who think global warming is very serious. Interretation of the -value (for this one-sided test): It s a conditional robability. Conditional on the null hyothesis being true (equal oulation roortions), what is the robability that we would observe a samle difference as large as the one observed or larger just by chance? Secific to this examle: If there really were no change in the roortion of the oulation who think global warming is very serious what is the robability of observing a samle roortion in 2009 that is.06 (6%) or more lower than the samle roortion in 2006? Answer: The robability is Therefore, we reject the idea (the hyothesis) that there was no change in the oulation roortion.