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Analysis of Environmental DataConceptual FoundationsConfidence Intervals and More

Topics:1. Population distribution of random variable2. Z-standardization of population distribution3. Sample distribution of random variable4. Z-standardization of sample distribution5. Sample estimate of population parameter6. Standard errors7. Confidence intervals

Primer on confidence intervals and more...Population distribution of a random variable

Population of fish

Y = fish size

This is not a confidence interval!

Primer on confidence intervals and more...Z standardization of population distribution

Primer on confidence intervals and more...Sample distribution of a random variable

Sample of fish

Y = fish size

A perfectlyrepresentativesample

This is not a confidence interval!

Primer on confidence intervals and more...Z standardization of sample distribution

Primer on confidence intervals and more...Sample estimate of population parameter

Sample of fish

Goal is to estimate

Our goal is to estimate the populationmean, ìy, from the sample y

OLS/MLE

Remember the sample mean is a random variablebecause it is derived from a random variable

Primer on confidence intervals and more...Many sample estimates of population parameter

Sample of fish

What if we could collect many samples (or even everypossible sample) and for each sample compute anestimate of the population parameter?

Primer on confidence intervals and more...Standard error of sample estimates of populationparameter

P If the distribution of thesample means is normal (andCLT says they always are), wecan calculate the variance andstandard deviation of thesample means – known as thestandard error

P But with only a single sample,we have to estimate thestandard error from our sample

Standard error of the mean:

Primer on confidence intervals and more...Standard error of sample estimates of populationparameter

Standard error of the mean:

sy = samplestandarddeviation

óy =populationstandarddeviation

Primer on confidence intervals and more...Standard error of sample estimates of populationparameter

Standard error of the mean:

P Tells us about the variation inour sample mean (underrepeated sampling)

P Tells us about the “error” inusing the sample mean toestimate the population mean

P Smaller variance in thepopulation and larger samplesize decrease the error in ourestimate

Primer on confidence intervals and more...Confidence interval for the sample estimate ofpopulation parameter

P Convert the distribution ofsample means into a standardnormal distribution via the z-score standardization

Confidence interval for the mean:

óy = populationstandarddeviation

This is a confidence interval!

Primer on confidence intervals and more...Confidence interval for the sample estimate ofpopulation parameter

Confidence interval for the mean:

This is a confidence interval!

P z variable (standard normal) iscalled a t statistic when we usethe sample estimate of the standarderror of the mean

sy = samplestandarddeviation

P t distribution is a symmetricalprobability distributioncentered around zero

P Similar to the normaldistribution except varies withsample size (actually degrees offreedom, n-1); has slightly fattertails than the normal butapproaches the normal when n(>30) is large

Primer on confidence intervals and more...Confidence interval for the sample estimate ofpopulation parameter

What is a t statistic?

P t distribution is a symmetricalprobability distributioncentered around zero

P Similar to the normaldistribution except varies withsample size (actually degrees offreedom, n-1); has slightly fattertails than the normal butapproaches the normal when n(>30) is large

Primer on confidence intervals and more...Confidence interval for the sample estimate ofpopulation parameter

What is a t statistic?

P z distribution is the probabilitydistribution of the z-standardized data or the z-standardized sample meansbased on population standarderror of mean

P t distribution is the probabilitydistribution for the z-standardized sample meansbased on sample estimate of thestandard error of mean

Primer on confidence intervals and more...Confidence interval for the sample estimate ofpopulation parameter

What is a t statistic?