1. Inference Concepts Confidence Regions

2. Inference ConceptsSlide #2 Confidence Regions What is the next logical question when H 0 is rejected?

3. 95% Confidence Interval Concept Confidence IntervalsSlide #3   se  se 95% of the samples provide a mean within what range?

4. Confidence IntervalsSlide #4   se  se Determine if  is contained in the constructed 95% CI Mark  x + 2SE -- this is a 95% CI Assume a sample results in the mean shown below YES

5. Confidence IntervalsSlide #5   se  se Repeat for more samples Sample #1 Sample #2 Sample #3 Sample #4 YES NO

6. Confidence IntervalsSlide #6   se  se So, how often will  be contained in  x + 2SE? What must happen for  to be contained in the  x+ 2SE interval? YES NO

7. Confidence IntervalsSlide #7  The technique provides a range that contains  95% of the time. 95% Confidence Interval Concept

8. Confidence IntervalsSlide #8  So, conclude that we are 95% confident* that  is contained in this CI. *95% of all 95% CIs will contain . 95% Confidence Interval Concept

9. Confidence IntervalsSlide #9 Not All CIs are 95% CIs, But the Concept Remains the Same The level of confidence used is 100(1-  )% Look at ciSim() C=0.90C=0.95C=0.99 n % ContnM.E. % ContnM.E. % ContnM.E. 10 905.2956.2998.1 30 903.0953.6994.7 60 902.1952.5993.3

10. Confidence IntervalsSlide #10 Constructing Any Confidence Region “Statistic” “sign” “margin-of-error” “Statistic” “sign” “constant” * SE ”statistic” General Specific

11. What is Z* Comes from a Z~N(0,1) The Z that has the level of confidence “in the direction” of the H A –Exact constant depends on direction of H A Inference ConceptsSlide #11

12. Inference ConceptsSlide #12 Confidence Regions if H A “not equals”, then ±Z* contains the level of confidence (C) interval likely contains the parameter if H A “less than”, then +Z* has C below upper bound for the parameter (parameter likely less) if H A “greater than”, then -Z* has C above lower bound for the parameter (parameter likely greater) C +z* C -z* C +z*-z*

13. Inference ConceptsSlide #13 Confidence Regions Suppose that  x=20, SE=3.5, and … –H A :  < ,  =0.05 95% confident that  is less than 25.76 –H A :  ≠ ,  =0.05 95% confident that  is between 13.14 & 26.86 –H A :  > ,  =0.01 99% confident that  is greater than 11.86 C +z* C -z* C