Hypothessis testing by Dr. O. Yusuf as part of the 5th Research Summer School - Jeddah at KAIMRC - WR

1. Sampling
Distributions, Standard
Error, Confidence Interval
Oyindamola Bidemi YUSUF
KAIMRC-WR

3. 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?

4. 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.

5. SAMPLING DISTRIBUTIONS
 Distribution of sample estimates
- Means
- Proportions
- Variance
 Take repeated samples and calculate
estimates
 Distribution is approximately normal

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. Sample 1 Mean
Sample 2 Mean
Sample 3 Mean
……….
….........
Sample n Mean
Standard error
Mean of the means
Mean of the means
This mean will also have a standard deviation= SE
Standard error

13. 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.

14. Interpreting SE
 Large SE indicates that estimate is
imprecise
 Small SE indicates that estimate is
precise
 How can SE be reduced?

15. Answer
 If sample size is increased
 If data is less variable

16. 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

17.  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

18. 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.

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

20. 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

21. 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

22. 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.

23.  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.

24. 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

25. 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.

26. Standard error = Standard Deviation
Square root of sample
size
14
√16

27.  95% CI: Statistic + 1.96 S.E. (Statistic)

28. ANWERS
 Standard error – 3.5
 95% confidence limits – 82.55 to 97.46

29. CI for a proportion
 P + 1.96 S.E. (P)
 SE(P)= √p(1-p)/n
 Online calculators are available

30. In summary
 SD versus SE
 Meaning and interpretation of CI
 Shopping for the right sampling
distribution

31.  THANK YOU