Inference Concepts

1. Inference Concepts Confidence Regions

2. Confidence Regions What is the next logical question when H0is rejected? Inference Concepts

3. 95% Confidence Interval Concept 95% of the samples provide a mean within what range? m m+2 se m-2 se Confidence Intervals

4. Assume a sample results in the mean shown below m m+2 se m-2 se Mark`x+ 2SE -- this is a 95% CI YES Determine if m is contained in the constructed 95% CI Confidence Intervals

5. Repeat for more samples m m+2 se m-2 se Sample #1 YES Sample #2 YES Sample #3 YES Sample #4 NO Confidence Intervals

6. What must happen for m to be contained in the`x+ 2SE interval? m m+2 se m-2 se So, how often will m be contained in `x +2SE? YES 95% YES YES NO Confidence Intervals

7. 95% Confidence Interval Concept The technique provides a range that contains m 95% of the time. m Confidence Intervals

8. Not All CIs are 95% CIs,But the Concept Remains the Same • The level of confidence used is 100(1-a)% • Look at ciSim() Confidence Intervals

9. General Specific Constructing Any Confidence Region “Statistic” “sign” “margin-of-error” “Statistic” “sign” “constant” * SE”statistic” Confidence Intervals

10. What is Z* • Comes from a Z~N(0,1) • The Z that has the level of confidence “in the direction” of the HA • Exact constant depends on direction of HA Inference Concepts

11. C -z* +z* C +z* C -z* Confidence Regions • if HA“not equals”, then ±Z*containsthe level of confidence (C) • interval likely contains the parameter • if HA“less than”, then +Z*has C below • upper bound for the parameter (parameter likely less) • if HA“greater than”, then -Z*has C above • lower bound for the parameter (parameter likely greater) Inference Concepts

12. C -z* +z* C +z* C -z* Confidence Regions • Suppose that `x=20, SE=3.5, and … • HA:m<28, a=0.05 • 95% confident that mis less than 25.76 • HA:m≠28, a=0.05 • 95% confident that m is between 13.14 & 26.86 • HA:m>11, a=0.01 • 99% confident that m is greater than 11.86 Inference Concepts