Download CI for Mean and Proportions

How sure we really are
Confidence intervals for means
and proportions
FETP India
Competency to be gained
from this lecture
Calculate 95% confidence intervals
for means and proportions
Key issues
• Concept of confidence interval
• Confidence interval for means
• Confidence interval for proportions
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
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
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
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
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
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
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
Confidence interval for a mean
• The confidence interval of the mean gives
the range of plausible values for the true
population mean
95%CI (x - 1.96

n
, x  1.96

n
)

Confidence interval for a mean
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”
10
10
95%CI (68 - 1.96
, 68  1.96
)
100
100
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
Calculating a 95% confidence interval
for a mean in practice
• Epi-Info, “Epitable” module
• Open-Epi calculator (Open source)
 www.openepi.com
• Excel
Confidence interval for a mean
Calculating a 95% confidence interval for
a mean in OpenEpi: 1/2 (Methods)
4. Click “calculate”
2. Click “Enter”
1. Choose “Mean, CI”
3. Enter data
Confidence interval for a mean
Calculating a 95% confidence interval for
a mean in OpenEpi: 1/2 (Results)
Confidence interval for a mean
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?
95%CI (37.5 - 1.96
12.2
350
, 37.5  1.96
12.1
350
)  (36,39)
Confidence interval for a mean
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
Using the binomial distribution
• The binomial distribution is a sampling
distribution for p
• Formula of the standard error:
SEproportion 
p(1 p)
n
 Where n = Sample size, p = proportion

Confidence interval for a proportion
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
Applying the concept of the confidence
interval of the mean to proportions
• For means, the 95% confidence interval
was:
95%CI (x - 1.96

n
, x  1.96

n
)
• 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
95%CI (p - 1.96
p(1 p)
p(1 p)
, p + 1.96
)
n
n
Confidence interval for a proportion
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
SE 
0.17(1 0.17)
0.17 x0.83

 0.019
363
363
• 95% confidence limits for the proportion are
0.17 1.96 x (0.019)

 Approximately 13% to 21%
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
Calculating a 95% confidence interval
for a proportion in practice
• Epi-Info, “Epitable” module
• Open-Epi calculator (Open source)
 www.openepi.com
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
Calculating a 95% confidence interval for
a proportion in OpenEpi: 1/2 (Results)
Confidence interval for a proportion
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?
0.46x0.54
0.46x0.54
95%CI (0.46 - 1.96
,0.46 + 1.96
)  (40,53)
250
250
Confidence interval for a proportion
From estimation to testing
• Confidence interval is about estimating
• The sampling distribution can also be used to
test hypotheses
 Statistical testing
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
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 +/- 1.92
the standard error, that is calculated using
different methods for means (s/√n) and
proportions (√[p(1-p)/n)]