Ch 11 – Ch 11 – Inference Inference

for for DistributioDistributio

nsnsYMS - 11.1YMS - 11.1 Inference Inference

for the for the Mean of a Mean of a PopulationPopulation

Basic Basic t(k)t(k)

One sample One sample tt-statistic-statistic– Uses Uses ss instead of σ and has instead of σ and has tt distribution distribution

with with n-1n-1 degrees of freedom degrees of freedom

Standard ErrorStandard Error– When standard deviation is estimatedWhen standard deviation is estimated

Compared to Standard Normal CurveCompared to Standard Normal Curve– Density curves are similar in shapeDensity curves are similar in shape– Spread is greater because substituting Spread is greater because substituting ss for σ for σ

introduces more variationintroduces more variation

As degrees of freedom As degrees of freedom change…change…

As As kk increases, increases, the the t(k)t(k) density density curve curve approaches approaches N(0,1) because N(0,1) because ss estimates σ more estimates σ more accurately when accurately when nn is large (last is large (last line of Table C in line of Table C in back of book) back of book)

p619 #11.1 to 11.7p619 #11.1 to 11.7

tt Intervals and Tests Intervals and Tests

Continue using toolbox for bothContinue using toolbox for both– Conditions and formulas are slightly different.Conditions and formulas are slightly different.– Conclusions are the same!Conclusions are the same!

When no row corresponds to When no row corresponds to kk degrees of degrees of freedom, round downfreedom, round down

p628 #11.9-11.11p628 #11.9-11.11

Two Sample vs.Two Sample vs.Matched Pairs Matched Pairs tt Procedures Procedures

Two samples (11.2)Two samples (11.2) Assumes samples are selected Assumes samples are selected

independently of each otherindependently of each other Matched pairsMatched pairs

Compare two treatments on same Compare two treatments on same subject by applying the one-sample subject by applying the one-sample tt procedures to the procedures to the observed differenceobserved difference

Ho: Ho: μμdiff = 0diff = 0

Last paragraph on p630-631Last paragraph on p630-631

Bull’s Eye ActivityBull’s Eye Activity

GuidelinesGuidelines Robust Procedures Robust Procedures

– When the confidence level or p-value When the confidence level or p-value doesn’t change much even when the doesn’t change much even when the assumptions of the procedure are violated assumptions of the procedure are violated

PowerPower– Ability to detect deviations from the null Ability to detect deviations from the null

hypothesis hypothesis – Go to Chapter 10 if you have questionsGo to Chapter 10 if you have questions

SRS is more important than population SRS is more important than population distribution being normal (except in distribution being normal (except in small samples)small samples)

Guidelines for Guidelines for nn

nn < 15 – if the data are close to < 15 – if the data are close to normalnormal

nn ≥ 15 – can be used except in the ≥ 15 – can be used except in the presence of outliers or strong presence of outliers or strong skewnessskewness

nn ≥ 40 – considered a large sample ≥ 40 – considered a large sample and and tt procedures can be used even procedures can be used even for clearly skewed distributions for clearly skewed distributions

p643 #11.27-11.28 CI examplesp643 #11.27-11.28 CI examplesp644 #11.31-11.32 graded group workp644 #11.31-11.32 graded group work

YMS - 11.2 YMS - 11.2

Comparing Two MeansComparing Two Means

Comparing the responses of two Comparing the responses of two treatments or characteristics of two treatments or characteristics of two populations when we populations when we have a separate have a separate sample from each sample from each treatment/populationtreatment/population

Need two independent SRSs and both Need two independent SRSs and both populations should be normally populations should be normally distributed distributed

p649 #11.37-11.38p649 #11.37-11.38

Two-Sample Problems Two-Sample Problems

RobustnessRobustness Two-sample procedures are more Two-sample procedures are more

robust than one-samplerobust than one-sample– Use previous guidelines for SUM of Use previous guidelines for SUM of

sample sizes (<15, 15 - 40, > 40)sample sizes (<15, 15 - 40, > 40)

Degrees of FreedomDegrees of Freedom Use smaller of two n-1 or let Use smaller of two n-1 or let

calculator find approximationcalculator find approximation

Approximates Degrees of Freedom Approximates Degrees of Freedom (accurate when sample sizes are both 5 or (accurate when sample sizes are both 5 or

larger) larger)

TI-83 for two-sample inference TI-83 for two-sample inference

Ch 7 Sampling Ch 7 Sampling Distribution ofDistribution of

Difference of the sample means is an Difference of the sample means is an unbiased estimator of the difference of the unbiased estimator of the difference of the population meanspopulation means

Variance of the difference is the sum of Variance of the difference is the sum of the variancesthe variances

If both population distributions are normal, If both population distributions are normal, then the distribution of the difference is then the distribution of the difference is also normal also normal

Two-sample Two-sample tt statistic statistic

Two-sample Two-sample tt interval interval

Pooled Procedures Pooled Procedures Assumes population variances are Assumes population variances are

the same and then averages the same and then averages (“pools”) them to estimate the (“pools”) them to estimate the common population variance common population variance

Don’t expect to do this…. ever.Don’t expect to do this…. ever.

o 11.2 Practice: p657 11.40-11.42, 11.47-11.2 Practice: p657 11.40-11.42, 11.47-11.48,11.50-11.52 11.48,11.50-11.52

o Graded group work: p670 11.56-11.58Graded group work: p670 11.56-11.58o Chapter Review: Stat Olympics and p675 Chapter Review: Stat Olympics and p675

#11.65-11.69, 11.73 odds#11.65-11.69, 11.73 odds