Inference Concepts

Confidence Regions

Inference Concepts Slide #2

Confidence Regions

What is the next logical question when H0 is rejected?

Confidence Intervals

95% Confidence Interval Concept

Slide #3

m m+2 sem-2 se

95% of the samples provide a mean within what range?

x

Confidence Intervals Slide #4

m m+2 sem-2 se

Determine if m is contained in the constructed 95% CI

x

Mark`x + 2SE -- this is a 95% CI

Assume a sample results in the mean shown below

YES

Confidence Intervals Slide #5

m m+2 sem-2 se

Repeat for more samples

Sample #1

Sample #2

Sample #3

Sample #4

YES

YES

YES

NO

Confidence Intervals Slide #6

m m+2 sem-2 seSo, how often will m be contained in `x + 2SE?

What must happen for m to be contained in the`x+ 2SE interval?

YES

YES

YES

NO

Confidence Intervals Slide #7

m

The technique provides a range that contains m 95% of the time.

95% Confidence Interval Concept

Confidence Intervals Slide #9

Not All CIs are 95% CIs,But the Concept Remains the Same

• The level of confidence used is 100(1-a)%

• Look at ciSim()

C=0.90 C=0.95 C=0.99

n%

Contn M.E.%

Contn M.E.%

Contn M.E.

10 90 5.2 95 6.2 99 8.130 90 3.0 95 3.6 99 4.760 90 2.1 95 2.5 99 3.3

Confidence Intervals Slide #10

Constructing Any Confidence Region

“Statistic” “sign” “margin-of-error”

“Statistic” “sign” “constant” * SE”statistic”

General

Specificn

zx*)//(

Inference Concepts

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

Slide #11

Inference Concepts Slide #12

Confidence Regions

• if HA “not equals”, then ±Z* contains the 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)

C+z*

C-z*

C

+z*-z*

Inference Concepts Slide #13

Confidence Regions• Suppose that `x=20, SE=3.5, and …

– HA:m<28, a=0.05• 95% confident that m is 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

C+z*

C

+z*-z*

C-z*