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

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

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

Reference population Target population Parameter Study population Representative Inference Sample Statistics

DESCRIPTIVE BIOSTATISTIC INFERENTIAL ESTIMATION SIGNIFANCE TESTING

Point Estimation Estimation Interval Estimation

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

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

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

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

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

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

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

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

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

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]

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

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)

Estimation & Probability 140 children had a mean urinary lead concentration of 2.18 with standard deviation 0.87. 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.

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

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 4.8 - 2.18 = 2.62. 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 2.576 but less than 3.291 in , so the probability of finding a deviation as large or more extreme than this lies between 0.01 and 0.001.

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)

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.

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

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

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

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

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

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

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

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

Confidence Intervals for the Population Mean µ

Example

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)

Example

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

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

Confidence Interval

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

CI of Proportions (P)

95% Confidence Interval for a Proportion

Example

Example…..

CI and Proportion

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.

Example Random sample of 59 students spent an average of Rm.273.20 on textbooks. Sample standard deviation was Rm.94.40. We can be 95% confident that the average amount spent by all students was between Rm.249.11 and Rm.297.29.

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?

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”.

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

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!!!)

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

Thanks