Download - RSS Hypothessis testing
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Sampling Distributions, Standard Error, Confidence Interval
Oyindamola Bidemi YUSUF KAIMRC-WR
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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?
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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.
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SAMPLING DISTRIBUTIONS Distribution of sample estimates
- Means- Proportions
- Variance Take repeated samples and
calculate estimates Distribution is approximately
normal
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Mathematicians have examined the distribution of these sample estimates and their results are expressed in the central limit theorem
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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
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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
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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
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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
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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
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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
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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.
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Interpreting SE
Large SE indicates that estimate is imprecise
Small SE indicates that estimate is precise
How can SE be reduced?
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Answer
If sample size is increased If data is less variable
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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
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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
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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.
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CONFIDENCE INTERVAL
Statistic + 1.96 S.E. (Statistic) 95% of the distribution of sample means lies within 1.96 SD of the population mean
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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
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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
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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.
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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.
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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
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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.
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Standard error = Standard Deviation Square root of sample size
14 √16
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95% CI: Statistic + 1.96 S.E. (Statistic)
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ANWERS
Standard error – 3.5 95% confidence limits – 82.55 to
97.46
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CI for a proportion
P + 1.96 S.E. (P) SE(P)= √p(1-p)/n Online calculators are available
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In summary
SD versus SE Meaning and interpretation of CI Shopping for the right sampling
distribution
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THANK YOU