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Ch 11 – Inference for Distributions YMS - 11.1 Inference for the Mean of a Population

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Basic t(k) One sample t-statistic One sample t-statistic –Uses s instead of σ and has t distribution with n-1 degrees of freedom Standard Error Standard Error –When standard deviation is estimated Compared to Standard Normal Curve Compared to Standard Normal Curve –Density curves are similar in shape –Spread is greater because substituting s for σ introduces more variation

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As degrees of freedom change… As k increases, the t(k) density curve approaches N(0,1) because s estimates σ more accurately when n is large (last line of Table C in back of book) As k increases, the t(k) density curve approaches N(0,1) because s estimates σ more accurately when n is large (last line of Table C in back of book) p619 #11.1 to 11.7

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t Intervals and Tests Continue using toolbox for both Continue using toolbox for both –Conditions and formulas are slightly different. –Conclusions are the same! When no row corresponds to k degrees of freedom, round down When no row corresponds to k degrees of freedom, round down p628 #11.9-11.11

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Two Sample vs. Matched Pairs t Procedures Two samples (11.2) Two samples (11.2) Assumes samples are selected independently of each other Assumes samples are selected independently of each other Matched pairs Matched pairs Compare two treatments on same subject by applying the one-sample t procedures to the observed difference Compare two treatments on same subject by applying the one-sample t procedures to the observed difference Ho: μdiff = 0 Ho: μdiff = 0 Last paragraph on p630-631 Bull’s Eye Activity

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Guidelines Robust Procedures Robust Procedures –When the confidence level or p-value doesn’t change much even when the assumptions of the procedure are violated Power Power –Ability to detect deviations from the null hypothesis –Go to Chapter 10 if you have questions SRS is more important than population distribution being normal (except in small samples) SRS is more important than population distribution being normal (except in small samples)

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Guidelines for n n < 15 – if the data are close to normal n < 15 – if the data are close to normal n ≥ 15 – can be used except in the presence of outliers or strong skewness n ≥ 15 – can be used except in the presence of outliers or strong skewness n ≥ 40 – considered a large sample and t procedures can be used even for clearly skewed distributions n ≥ 40 – considered a large sample and t procedures can be used even for clearly skewed distributions p643 #11.27-11.28 CI examples p644 #11.31-11.32 graded group work

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YMS - 11.2 Comparing Two Means

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Comparing the responses of two treatments or characteristics of two populations when we have a separate sample from each treatment/population Comparing the responses of two treatments or characteristics of two populations when we have a separate sample from each treatment/population Need two independent SRSs and both populations should be normally distributed Need two independent SRSs and both populations should be normally distributed p649 #11.37-11.38 Two-Sample Problems

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Robustness Two-sample procedures are more robust than one-sample Two-sample procedures are more robust than one-sample –Use previous guidelines for SUM of sample sizes ( 40) Degrees of Freedom Use smaller of two n-1 or let calculator find approximation Use smaller of two n-1 or let calculator find approximation

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Approximates Degrees of Freedom (accurate when sample sizes are both 5 or larger) TI-83 for two-sample inference

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Ch 7 Sampling Distribution of Difference of the sample means is an unbiased estimator of the difference of the population means Difference of the sample means is an unbiased estimator of the difference of the population means Variance of the difference is the sum of the variances Variance of the difference is the sum of the variances If both population distributions are normal, then the distribution of the difference is also normal If both population distributions are normal, then the distribution of the difference is also normal

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Two-sample t statistic Two-sample t interval

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Pooled Procedures Assumes population variances are the same and then averages (“pools”) them to estimate the common population variance Assumes population variances are the same and then averages (“pools”) them to estimate the 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- 11.48,11.50-11.52 o Graded group work: p670 11.56-11.58 o Chapter Review: Stat Olympics and p675 #11.65-11.69, 11.73 odds

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