1. Confidence intervals for means and proportions FETP India
How sure we really are
Confidence intervals for means and proportions FETP India

2. Competency to be gained from this lecture
Calculate 95% confidence intervals for means and proportions

3. Key issues Concept of confidence interval
Confidence interval for means
Confidence interval for proportions

4. What we learnt so far (1/3)
Population parameters are fixed
We can take samples from the population
Several samples of size ‘n’ are possible
Each sample give estimates (e.g., means) called “statistics”
Statistics vary from sample to sample
This is called “Sampling fluctuation”
Concept of confidence interval

5. What we learnt so far (2/3)
The distribution of a statistic for all possible samples of given size ‘n’ is called “sampling distribution”
For large ‘n’, the sampling distribution is ‘normal’ even if the original distribution is not
If the original distribution is normal, the result is true even for small ‘n’
Concept of confidence interval

6. What we learnt so far (3/3)
The mean of the sampling distribution is the ‘population mean’
The standard deviation of the sampling distribution is known as standard error
SE= Population SD /√n
Concept of confidence interval

7. Concept of confidence interval
Easy to estimate the standard deviation, difficult to estimate the mean
Samples generate sample means and standard error
The usefulness of these parameters vary:
The standard deviation from a single sample as an estimate of population SD for large ‘n’ is fair
The mean from a single sample as an estimate of population mean may not be
Concept of confidence interval

8. How can the population mean be estimated?
It is desirable to give a range of values with a specific level of confidence that the true population mean is one of the values in the range
We can obtain this using the sampling distribution – which is ‘normal’ using the properties of ‘normal’ distribution
Mean
Standard deviation
Concept of confidence interval

9. From the standard error (SE) to the confidence interval
The point estimate x (mean in the sample) is a point in the sampling distribution and there is a 95% chance that it lies in the µ1.96 SE interval
But µ is not known
Interchanging µ and x we can infer that there is a 95% chance that µ lies in the interval x 1.96 SE
Concept of confidence interval

10. Inference using various levels of confidence
Using the properties of the normal distribution, we can infer what proportion of the values lie between values
Considering the distribution of the means:
68% of sample means will lie within 1 standard deviation above or below the sample mean
95% of sample means will lie within 1.96 standard deviation above or below the sample mean
“1.96” come from the standard z table for alpha=0.05
Concept of confidence interval

11. Confidence interval for a mean
The confidence interval of the mean gives the range of plausible values for the true population mean
Confidence interval for a mean

12. Example of a calculation of a confidence interval for a mean
Sample of 100 observations,
Mean height is 68”
SD: 10”
Standard error of the mean = 10 /  100 = 1
95% confidence limits for population mean are 68  1.96 x (1)
Approximately 66” to 70”
Confidence interval for a mean

13. Confidence interval for a mean
Interpretation of the calculation of the confidence interval for a mean
The 95% confidence interval for the mean of 68 is (66, 70)
This means that with repeated random sampling, 95% of the intervals will contain the true mean (µ)
Since we have one of these intervals, we can be 95% confident that this interval contains the true mean
Confidence interval for a mean

14. Calculating a 95% confidence interval for a mean in practice
Epi-Info, “Epitable” module
Open-Epi calculator (Open source)
Excel
Confidence interval for a mean

15. Confidence interval for a mean
Calculating a 95% confidence interval for a mean in OpenEpi: 1/2 (Methods)
4. Click “calculate”
2. Click “Enter”
3. Enter data
1. Choose “Mean, CI”
Confidence interval for a mean

16. Confidence interval for a mean
Calculating a 95% confidence interval for a mean in OpenEpi: 1/2 (Results)
Confidence interval for a mean

17. Exercise to calculate the 95% confidence interval for a mean
Study of gestational age at birth in the past month in a sample of health care facilities
Results of the study
n=350 births
Sample mean= 37.5 weeks
s=12.2
What is the 95% confidence interval?
Confidence interval for a mean

18. Confidence interval for a proportion
Applying the same methods to generate confidence intervals for proportions
The central limit theorem also applies to distribution of sample proportions when the sample size is large enough
The population proportion replaces the population mean
The binomial distribution replaces the normal distribution
Confidence interval for a proportion

19. Using the binomial distribution
The binomial distribution is a sampling distribution for p
Formula of the standard error:
Where n = Sample size, p = proportion
Confidence interval for a proportion

20. Using the central limit theorem
As the sample n increases, the binomial distribution becomes very close to a normal distribution (Central limit theorem)
Thus, we can use the normal distribution to calculate confidence intervals and test hypotheses
If np and n (1-p) and equal to 10 or more, then the normal approximation may be used
Confidence interval for a proportion

21. Confidence interval for a proportion
Applying the concept of the confidence interval of the mean to proportions
For means, the 95% confidence interval was:
For proportions, we just replace the formula of the standard error of the mean by the standard error of the proportion that comes from the binomial distribution
Confidence interval for a proportion

22. Calculation of a confidence interval for a proportion: Prevalence of goiter in Solan, Himachal Pradesh, India, 2005
Sample of 363 children:
63 (17%) present with goiter
Standard error of the proportion
95% confidence limits for the proportion are  1.96 x (0.019)
Approximately 13% to 21%

23. Confidence interval for a proportion
Interpretation of the calculation of the confidence interval for the proportion
The 95% confidence interval for the proportion of 17% is (13%, 21%)
This means that with repeated random sampling, 95% of the intervals will contain the true proportion
Since we have one of these intervals, we can be 95% confident that this interval contains the true proportion
Confidence interval for a proportion

24. Calculating a 95% confidence interval for a proportion in practice
Epi-Info, “Epitable” module
Open-Epi calculator (Open source)
Confidence interval for a proportion

25. Confidence interval for a proportion
Calculating a 95% confidence interval for a proportion in OpenEpi: 1/2 (Methods)
2. Click “Enter”
4. Click “calculate”
1. Choose “Proportion”
3. Enter data
Confidence interval for a proportion

26. Confidence interval for a proportion
Calculating a 95% confidence interval for a proportion in OpenEpi: 1/2 (Results)
Confidence interval for a proportion

27. Exercise to calculate the 95% confidence interval for a proportion
In a sample of 250 HIV infected persons with AIDS, 116 are positive for tuberculosis
What is the 95% confidence interval?
Confidence interval for a proportion

28. From estimation to testing
Confidence interval is about estimating
The sampling distribution can also be used to test hypotheses
Statistical testing

29. Dealing with non-normal parent population
If sample size exceeds 30, we are safe because the sampling distribution will approach the normal distribution
If the sample size is smaller than 30, the distribution is different
The 1.96 value will be replaced by another value coming from the t-distribution
Slightly different from the normal distribution
Depends upon the sample size
The degrees of value will be n-1

30. Take home messages
Confidence intervals use the central limit theorem to estimate a range of possible values for the population parameter on the basis of the sample estimate, the standard deviation and the sample size
The 95% confidence intervals lies at +/ the standard error, that is calculated using different methods for means (s/√n) and proportions (√[p(1-p)/n)]