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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?