Sampling Distributions, Standard Error, Confidence Interval

Oyindamola Bidemi YUSUF KAIMRC-WR

SAMPLE

Why do we sample? Note: information in sample may

not fully reflect what is true in the population

We have introduced sampling error by studying only some of the population

Can we quantify this error?

SAMPLING VARIATIONS Taking repeated samples Unlikely that the estimates would be

exactly the same in each sample However, they should be close to the

true value By quantifying the variability of these

estimates, precision of estimate is obtained.

Sampling error is thereby assessed.

SAMPLING DISTRIBUTIONS Distribution of sample estimates

- Means- Proportions

- Variance Take repeated samples and

calculate estimates Distribution is approximately

normal

Mathematicians have examined the distribution of these sample estimates and their results are expressed in the central limit theorem

central limit theorem

Sampling distributions are approximately normally distributed regardless of the nature of the variable in the parent population

The mean of the sampling distribution is equal to the true population mean Mean of sample means is an unbiased

estimate of the true population mean The standard deviation (SD) of sampling

distribution is directly proportional to the population SD and inversely proportional to the square root of the sample size

SUMMARY: DISTRIBUTION OF SAMPLE ESTIMATES

NORMAL Mean = True population mean Standard deviation = Population

standard deviation divided by square root of sample size

Standard deviation called standard error

ESTIMATION A major purpose or objective of health

research is to estimate certain population characteristics or phenomena

Characteristic or phenomenon can be quantitative such as average SYSTOLIC BLOOD PRESSURE of adult men or qualitative such as proportion with MALNUTRITION

Can be POINT or INTERVAL ESTIMATE

Point estimates

Value of a parameter in a population e.g. mean or a proportion

We estimate value of a parameter using using data collected from a sample

This estimate is called sample statistic and is a POINT ESTIMATE of the parameter i.e. it takes a single value

STANDARD ERROR Used to describe the variability of

sample means Depends on variability of individual

observations and the sample size Relationship described as –Standard error = Standard Deviation

Square root of sample size

Sample 1 Mean

Sample 2 Mean

Sample 3 Mean

……….

….........

Sample n Mean

Standard error

Mean of the means

Mean of the meansThis mean will also have a standard deviation= SE

Standard error

Standard Deviation or Standard Error?

Quote standard deviation if interest is in the variability of individuals as regards the level of the factor being investigated – SBP, Age and cholesterol level.

Quote standard Error if emphasis is on the estimate of a population parameter.It is a measure of uncertainty in the sample statistic as an estimate of population parameter.

Interpreting SE

Large SE indicates that estimate is imprecise

Small SE indicates that estimate is precise

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 of precision into an interval estimate for the population parameter

How? By using the knowledge of the theoretical

probability distribution of the sample statistic to calculate a CI

Not sufficient to rely on a single estimate

Other samples could yield plausible estimates

Comfortable to find a range of values within which to find all possible mean values

WHAT IS A CONFIDENCE INTERVAL?

The CI is a range of values, above and below a finding, in which the actual value is likely to fall.

The confidence interval represents the accuracy or precision of an estimate.

Only by convention that the 95% confidence level is commonly chosen.

Researchers are confident that if other surveys had been done, then 95 per cent of the time — or 19 times out of 20 — the findings would fall in this range.

CONFIDENCE INTERVAL

Statistic + 1.96 S.E. (Statistic) 95% of the distribution of sample means lies within 1.96 SD of the population mean

Interpretation

If experiment is repeated many times, the interval would contain the true population mean on 95% of occasions

i.e. a range of values within which we are 95% certain that the true population mean lies

Issues in CI interpretation

How wide is it? A wide CI indicates that estimate is imprecise

A narrow one indicates a precise estimate

Width is dependent on size of SE, which in turn depends on SS

Factors affecting CI A narrow or small confidence interval

indicates that if we were to ask the same question of a different sample, we are reasonably sure we would get a similar result.

A wide confidence interval indicates that we are less sure and perhaps information needs to be collected from a larger number of people to increase our confidence.

Confidence intervals are influenced by the number of people that are being surveyed.

Typically, larger surveys will produce estimates with smaller confidence intervals compared to smaller surveys.

Why are CIs important

Because confidence intervals represent the range of values scores that are likely if we were to repeat the survey.

Important to consider when generalizing results.

Consider random sampling and application of correct statistical test

Like comfort zones that encompass the true population parameter

Calculating confidence limits

The mean diastolic blood pressure from 16 subjects is 90.0 mm Hg, and the standard deviation is 14 mm Hg. Calculate its standard error and 95% confidence limits.

Standard error = Standard Deviation Square root of sample size

14 √16

95% CI: Statistic + 1.96 S.E. (Statistic)

ANWERS

Standard error – 3.5 95% confidence limits – 82.55 to

97.46

CI for a proportion

P + 1.96 S.E. (P) SE(P)= √p(1-p)/n Online calculators are available

In summary

SD versus SE Meaning and interpretation of CI Shopping for the right sampling

distribution

THANK YOU