SAMPLE Why do we sample? Note: information in sample may notfully reflect what is true in thepopulation We have introduced sampling error bystudying only some of the population Can we quantify this error?
SAMPLING VARIATIONS Taking repeated samples Unlikely that the estimates would be exactlythe same in each sample However, they should be close to the truevalue By quantifying the variability of theseestimates, precision of estimate is obtained. Sampling error is thereby assessed.
SAMPLING DISTRIBUTIONS Distribution of sample estimates- Means- Proportions- Variance Take repeated samples and calculateestimates Distribution is approximately normal
Mathematicians have examined thedistribution of these sample estimatesand their results are expressed in thecentral limit theorem
central limit theorem Sampling distributions are approximately normallydistributed regardless of the nature of the variable inthe parent population The mean of the sampling distribution is equal to thetrue population mean Mean of sample means is an unbiased estimate ofthe true population mean The standard deviation (SD) of sampling distributionis directly proportional to the population SD andinversely proportional to the square root of thesample size
SUMMARY: DISTRIBUTION OFSAMPLE ESTIMATES NORMAL Mean = True population mean Standard deviation = Population standarddeviation divided by square root of samplesize Standard deviation called standard error
ESTIMATION A major purpose or objective of healthresearch is to estimate certain populationcharacteristics or phenomena Characteristic or phenomenon can bequantitative such as average SYSTOLICBLOOD PRESSURE of adult men or qualitativesuch as proportion with MALNUTRITION Can be POINT or INTERVAL ESTIMATE
Point estimates Value of a parameter in a populatione.g. mean or a proportion We estimate value of a parameter usingusing data collected from a sample This estimate is called sample statisticand is a POINT ESTIMATE of theparameter i.e. it takes a single value
STANDARD ERROR Used to describe the variability ofsample means Depends on variability of individualobservations and the sample size Relationship described as –Standard error = Standard DeviationSquare root of samplesize
Sample 1 MeanSample 2 MeanSample 3 Mean……….….........Sample n MeanStandard errorMean of the meansMean of the meansThis mean will also have a standard deviation= SEStandard error
Standard Deviation or StandardError? Quote standard deviation if interest is in thevariability of individuals as regards the levelof the factor being investigated – SBP, Ageand cholesterol level. Quote standard Error if emphasis is on theestimate of a population parameter.It is a measure of uncertainty in the samplestatistic as an estimate of populationparameter.
Interpreting SE Large SE indicates that estimate isimprecise Small SE indicates that estimate isprecise How can SE be reduced?
Answer If sample size is increased If data is less variable
INTERVAL ESTIMATE Is SE particularly useful? More helpful to incorporate this measure ofprecision into an interval estimate for thepopulation parameter How? By using the knowledge of the theoreticalprobability distribution of the sample statistic tocalculate a CI
Not sufficient to rely on a singleestimate Other samples could yield plausibleestimates Comfortable to find a range of valueswithin which to find all possible meanvalues
WHAT IS A CONFIDENCEINTERVAL? The CI is a range of values, above and below afinding, in which the actual value is likely to fall. The confidence interval represents the accuracy orprecision of an estimate. Only by convention that the 95% confidence levelis commonly chosen. Researchers are confident that if other surveyshad been done, then 95 per cent of the time — or19 times out of 20 — the findings would fall in thisrange.
CONFIDENCE INTERVAL Statistic + 1.96 S.E. (Statistic) 95% of the distribution of samplemeans lies within 1.96 SD of thepopulation mean
Interpretation If experiment is repeated many times,the interval would contain the truepopulation mean on 95% of occasions i.e. a range of values within which weare 95% certain that the truepopulation mean lies
Issues in CI interpretation How wide is it? A wide CI indicates thatestimate is imprecise A narrow one indicates a preciseestimate Width is dependent on size of SE, whichin turn depends on SS
Factors affecting CI A narrow or small confidence intervalindicates that if we were to ask the samequestion of a different sample, we arereasonably sure we would get a similar result. A wide confidence interval indicates that weare less sure and perhaps information needsto be collected from a larger number ofpeople to increase our confidence.
Confidence intervals are influenced bythe number of people that are beingsurveyed. Typically, larger surveys will produceestimates with smaller confidenceintervals compared to smaller surveys.
Why are CIs important Because confidence intervals representthe range of values scores that arelikely if we were to repeat the survey. Important to consider whengeneralizing results. Consider random sampling andapplication of correct statistical test Like comfort zones that encompass thetrue population parameter
Calculating confidence limits The mean diastolic blood pressure from16 subjects is 90.0 mm Hg, and thestandard deviation is 14 mm Hg.Calculate its standard error and 95%confidence limits.
Standard error = Standard DeviationSquare root of samplesize14√16