1. Dr.Theingi Community Medicine
CONFIDENCE INTERVAL
Dr.Theingi
Community Medicine

2. Confidence Interval Learning objectives of this lecture are to
Understand the Approaches to Statistical Inference:
1. Estimating Parameters
2. Hypothesis testing
Discuss Construction of Confidence Intervals
Know Standard Errors
Learn the Calculation and interpretation of confidence interval

3. Confidence Interval Learning outcomes
At the end of this lecture the students should be able to
Describe the two approaches to Statistical Inference:
1. Estimating Parameters 2. Hypothesis testing
Explain Standard Errors
Calculate and interpret confidence interval
Explain Significance & uses of Confidence interval

4. Reference population
Target population
Parameter
Study population
Representative
Inference
Sample
Statistics

5. DESCRIPTIVE
BIOSTATISTIC
INFERENTIAL
ESTIMATION
SIGNIFANCE TESTING

6. Point Estimation
Estimation
Interval Estimation

7. Two approaches to estimate population parameters
Point estimation:
Obtain a value estimate for the population parameter
Interval estimation:
Construct an interval within which the population parameter will lie with a certain probability

8. Point estimate Provides single value
Based on observations from 1 sample
Gives no information about how close value is to the unknown population parameter
Point estimates of population parameters are prone to sampling error 19

9. Point estimate Example:
Sample mean is a good estimate of the population mean
Sample proportion

10. Interval Estimation Provides range of values
Based on observations from 1 sample
Gives information about closeness to unknown population parameter
Stated in terms of probability
Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter; A 95% confidence level means that 95% of the intervals would include the parameter; and so on. 22

11. Interval Estimation
An interval estimator is a formula that determines, a range within which the population parameter lies with certain probability
Example: Unknown population mean lies between 50 & 70 with 95% confidence

12. Estimation Process
Population
Mean, m, is unknown J J J J J J J J J 7

13. Estimation Process J J J J J J J Random Sample J J J J Population
Mean `X = 30 J
Mean, m, is unknown J J J J J J
Sample J J J J 8

14. I am 95% confident that m is between 20 & 40.
Estimation Process
Population
Random Sample
I am 95% confident that m is between 20 & 40.
Mean `X = 30 J
Mean, m, is unknown J J J J J J
Sample J J J J 9

15. Interval Estimation
To indicate this Variability they use Interval estimates  are called CONFIDENCE INTERVALS

16. Confidence Intervals
The confidence interval is the range in which we expect the population mean to fall
The confidence level is the degree of assurance that our mean accurately represents the population mean (e.g 95%, 99%)
How do we calculate?
Point Estimate  [Reliability Factor  Standard Error]

17. Constructing Confidence Intervals
Confidence intervals have similar structures
Point Estimate  [Reliability Factor  Standard Error]
Reliability factor is a number based on the assumed distribution of the point estimate and the level of confidence
Standard error of the sample statistic

18. Sampling Distribution of the Mean
Normal Population Distribution
σ is the
Population
Standard
Deviation
σ Xbar is the
Sample Standard
Deviation.
σ Xbar = σ/√n
σ Xbar << σ
SE<<SD
Normal Sampling Distribution
(has the same mean)

19. Estimation & Probability
140 children had a mean urinary lead concentration of 2.18 with standard deviation The points that include 95% of the observations are 2.18 ± (1.96*0.87), giving a range of 0.48 to 3.89.
This observation is greater than 3.89 and so falls in the 5% beyond the 95% probability limits. We can say that the probability of each of such observations occurring is 5% or less.

20. Another way of looking at this is to see that if one chose one child at random out of the 140, the chance that their urinary lead concentration exceeded 3.89 or was less than 0.48 is 5%.
Standard deviations thus set limits about which probability statements can be made

21. to estimate the probability of finding an observed value, say a urinary lead concentration of 4.8 , in sampling from the same population of observations as the 140 children provided
The distance of the new observation from the mean is = How many standard deviations does this represent? Dividing the difference by the standard deviation gives 2.62/0.87 = 3.01.
This number is greater than but less than in , so the probability of finding a deviation as large or more extreme than this lies between 0.01 and

22. Z Distribution
A transformation of normal distributions into a standard form with a mean of 0 and a standard deviation of 1. It is sometimes useful.
μ = 8
σ = 10
μ = 0
σ = 1 X Z 8
8.6
0.12
P(X < 8.6)
P(Z < 0.12)

23. Calculation
• If your C.L is 95%, calculated “Z” value has to be compared with 1.96 which is the “Z” value for specified 95% C.L.

24. 140 children had a mean urinary lead concentration of 2
140 children had a mean urinary lead concentration of 2.18 with standard deviation Estimate the probability of finding an observed value, say a urinary lead concentration of 4.8
Z= / 0.87√140
z=3.01
Probability= 0.001

26. CONFIDENCE LEVEL
The confidence level is the degree of assurance that our mean accurately represents the population mean.
Typical values are 99%, 95%, 90%
Probability that the unknown population parameter falls within interval
Denoted (1 - a) %
a is probability that parameter is not within interval 34

27. Confidence Interval if δ is Known
Using X
X units:
Lower
Confidence
Limit
Xmin
Point Estimate for Xbar
Upper
Confidence
Limit
Xmax

28. CONFIDENCE LIMITS The ends of the CI are called confidence limits
Example
95% of the observations are within the range of 0.48 to 3.89

29. Two interpretations of confidence intervals
In the probabilistic interpretation, we say that
A 95% confidence interval for a population parameter means that, in repeated sampling, 95% of such confidence intervals will include the population parameter
In the practical interpretation, we say that
We are 95% confident that the 95% confidence interval will include the population parameter

30. CI’s can be established for any population parameters
Example: C I for
Mean
Proportion
Relative risks
Odds ratio
Correlation
Difference between 2 means,
Difference between 2 proportions etc

31. standard error
Statisticians have derived formulas to calculate the standard deviation of the sampling distribution [distribution of the sample means]
It’s called the standard error of the statistic
Calculating Standard Error of the Mean
SD- describes variation in data values. SE-describes the precision of the sample mean.
If sample size increases---SE decreases

32. Standard Errors: 2 types
1.– Standard error of the mean (SEM)
2.– Standard error of a proportion (SEP)
They always involve the sample size ‘n’
As the sample size gets bigger,
the standard error gets smaller

33. Confidence Intervals for the Population Mean µ

34. Example

35. Confidence Interval
For a 99% CI  ± 2.6 SEM , For a 95% CI ± 2 SEM , For a 90% CI± 1.65 SE
About 95% of the time, the sample mean (or proportion) will be within two standard errors [2SE] of the population mean (or proportion)

36. Example

37. Ways to Write a Confidence Interval of 125+/-2.8

38. Are all CIs 95%? -No It is the most commonly used
-A 99% CI is wider
-A 90% CI is narrower
To be “more confident” you need a
bigger interval
– For a 99% CI, you need ± 2.6 SEM
– For a 95% CI, you need ± 2 SEM
– For a 90% CI, you need ± 1.65 SEM

39. Confidence Interval

40. Interpretation
If we were to take 100 random samples each of the same size, approximately 95 of the CIs would include the true value of pop.µ

41. CI of Proportions (P)

42. 95% Confidence Interval for a Proportion

43. Example

44. Example…..

45. CI and Proportion

46. As long as you have a “large” sample….
A confidence interval for a population mean is:
where the average, standard deviation, and n depend on the sample, and Z depends on the confidence level.

47. Example
Random sample of 59 students spent an average of Rm on textbooks. Sample standard deviation was Rm
We can be 95% confident that the average amount spent by all students was between Rm and Rm

48. What happens if you can only take a “small” sample?
Random sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.
What is the average amount all students slept last night?

49. If you have a “small” sample...
Replace the Z value with a t value to get:
where “t” comes from Student’s t distribution, and depends on the sample size through the degrees of freedom “n-1”.

50. Let’s get back to our example!
Sample of 15 students slept an average of 6.4 hours last night with standard deviation of 1 hour.
Need t with n-1 = 15-1 = 14 d.f.
t table value for 95% confidence, t14 = 2.145

53. That is...
We can be 95% confident that average amount slept last night by all students is between 5.85 and 6.95 hours.
But! Adults need 8 hours of sleep each night.
Logical conclusion: Students need more sleep.
(Just don’t get it in this class!!!)

54. What happens to CI as sample gets larger?
For large samples:
Z and t values become almost identical,
so CIs are almost identical.

55. Thanks