Inferences about two Populations Dependent vs Independent samples
Inferences about two Populations Dependent vs. Independent samples
Independent and Dependent Samples Two samples are independent if the sample selected from one population is not related to the sample selected from the other population. If one sample is related to the other, the samples are dependent. With dependent samples we get two values for each person, sometimes called paired-samples.
Questions in Statistical Notation In general we compare two populations by seeing how their difference compares to 0.
Estimator In both cases we can use as a point estimate of: This can then be then used to tell if the means are the same, different or if one is bigger than the other using the distribution of , hypothesis testing and confidence intervals.
Dependent samples With dependent samples we take advantage of the fact that we have two values for each person and take the difference, d, of these. This forms a data set with which we can find a confidence interval or hypothesis test or the difference between the population means.
Example
Independent large samples With independent samples things are slightly more complicated. Sometimes they form sets which are not the same size. Here we can work as before with a normal distribution (z), but we need an appropriate formula for the standard deviation. Here it is:
Independent large samples With independent samples things are slightly more complicated. Sometimes they form sets which are not the same size. Here we can work as before with a normal distribution (z), but we need an appropriate formula for the standard deviation. Here it is:
Independent small samples With small independent samples things are again slightly more complicated. We use the same standard deviation as with big samples, but to estimate the standard deviation of population 1 and population 2 we assume they are the same and get a “pooled” standard deviation.
Independent small samples With small independent samples things are again slightly more complicated. We use the same standard deviation as with big samples, but to estimate the standard deviation of population 1 and population 2 we assume they are the same and get a “pooled” standard deviation.
Example
Two Population Proportions We can use as a point estimate of: This can then be then used to tell if the proportions are the same, different or if one is bigger than the other using the distribution of , hypothesis testing and confidence intervals.
Two Population Proportions With randomly selected, independent samples we can proceed as before with a normal distribution (z), but we need an appropriate formula for the standard deviation. Here it is:
Example
Problems • • • #9. 36 on page 465 #9. 42 on page 466 #9. 6 on page 448 #9. 14 on page 449 #9. 58 on page 477 #9. 60 on page 477
Overview Comparing Two Populations: – Mean (Small Dependent Samples) – Mean (Large Independent Samples) – Proportion
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