Ch 8 Estimating with Ch 8 Estimating with ConfidenceConfidence

8.1: Confidence Intervals 8.1: Confidence Intervals

DefinitionsDefinitions Statistical Inference Statistical Inference

– Provides methods for drawing Provides methods for drawing conclusions about a population from conclusions about a population from sample datasample data

– What happens if I do this many times?What happens if I do this many times?

Formal InferenceFormal Inference– Using probability to express the strength Using probability to express the strength

of our conclusions (probability takes of our conclusions (probability takes chance variation into account)chance variation into account)

Margin of Error Margin of Error

– How accurate we How accurate we believe our believe our guess is, based guess is, based on the variability on the variability of the estimateof the estimate

– What you always What you always see in the fine see in the fine printprint

Confidence Intervals Confidence Intervals Use facts about sampling distributions (what Use facts about sampling distributions (what

would happen in the long run) to express our would happen in the long run) to express our confidence in the results of any one sampleconfidence in the results of any one sample

““We got these numbers by a method that gives We got these numbers by a method that gives correct results 95% of the time.”correct results 95% of the time.”

In the form of In the form of estimate ± margin of errorestimate ± margin of error

A confidence level, C, gives the overall success A confidence level, C, gives the overall success rate of the method for calculating the rate of the method for calculating the confidence interval. In C% of all possible confidence interval. In C% of all possible samples, the method would yield an interval samples, the method would yield an interval that captures the true parameter value.that captures the true parameter value.

Most Common Confidence Most Common Confidence IntervalsIntervals

90% corresponds to the middle 90% of normal distribution z-score = 1.645 (called the critical value)

95% corresponds to the middle 95% of normal distribution z-score = 1.960

99% corresponds to the middle 99% of normal distribution z-score = 2.576

Interpretation: We are C% confident that the interval from _____ to ____captures the actual value of the population parameter in context.

Ways to Decrease Margin of Ways to Decrease Margin of Error Error

Make z* smaller (decrease confidence level Make z* smaller (decrease confidence level C)C)

Increase sample size Increase sample size nn

Decrease standard deviation Decrease standard deviation σσ

– High Confidence – our method almost always High Confidence – our method almost always gives correct answersgives correct answers

– Small Margin of Error – we have pinned down the Small Margin of Error – we have pinned down the parameter quite precisely parameter quite precisely

Confidence Intervals Confidence Intervals Toolbox Toolbox II CCreate reate FFabulous abulous CConfidence onfidence IIntervalsntervals

Identify Identify – PopulationPopulation– ParameterParameter– ProcedureProcedure

ConditionsConditions FormulaFormula CalculationsCalculations Interpret in Interpret in

ContextContext

Many CIs Collect More Measures