1. Chapter 3 INTERVAL ESTIMATES
BAE 5333 Applied Water Resources Statistics
Biosystems and Agricultural Engineering Department
Division of Agricultural Sciences and Natural Resources
Oklahoma State University
Source
Dr. Dennis R. Helsel & Dr. Edward J. Gilroy
2006 Applied Environmental Statistics Workshop
and
Statistical Methods in Water Resources

2. Population vs. Sample
We measure characteristics of a sample and infer that they apply to the population.

3. Intervals
An interval computed from sample data provides information on how certain we are of the true population parameter.

4. Interval Estimates
Confidence Interval – contains an unknown parameter (mean, median) of the population with a specified probability
Prediction Interval – contains one or more future observations with a specified probability
Tolerance Interval – contains a proportion (percentile) of future observations with a specified probability

5. What is Inside the Interval?
Confidence Interval – contains an unknown parameter (mean, median) of the population with a specified probability
Prediction Interval – contains one or more future observations with a specified probability
Tolerance Interval – contains a proportion (percentile) of future observations with a specified probability

6. Your Interval May Not
Contain the True Value!

7. Meaning of a Confidence Interval
If you compute ten 90% confidence intervals
Each from a different sample of data collected under identical conditions with identical methods
Thus, each sample is equally valid
Nine of the 10 intervals (90%) will contain the true mean.
One will not!
You never know if yours is that one!!!

8. Meaning of a Confidence Interval Ten 90% Confidence Intervals

9. Meaning of a Confidence Interval
Example:
90% Confidence Interval about the Mean
We are 90% confident that the true mean turbidity in the Poteau River is between
5 and 200 NTU (Nephelometric Turbidity Units).

10. Computing Confidence Intervals
Parametric Intervals
μ = population mean
X = sample mean
z = depends on confidence level
σ = standard error of the mean _
Symmetric around the sample mean
Confidence levels are valid if data are normally distributed or there are a large amount of data

11. Computing Confidence Intervals Nonparametric Intervals
Usually computed on median or other percentile
Endpoints are data values
Count the same number of data observations from each end of the ranked dataset
Does not depend on assumption that data are normally distributed

12. Confidence Intervals on Skewed Data
Parametric intervals assume data follow a normal distribution or the mean does.
If this is incorrect, the confidence intervals will not include the true value as often as the confidence interval suggests.

13. Confidence Intervals on Skewed Data
First Approach
Transform data to approximate normality
Compute the confidence interval
Problem
When the confidence interval is retransformed back to the original units, it is no longer a confidence internal on the mean (i.e. the confidence interval is only valid in transformed space)
With logs, it is a confidence interval on the geometric mean; an estimate of the median

14. Confidence Intervals on Skewed Data
Example
Arsenic Concentrations New Hampshire Ground Water

15. Confidence Intervals on Skewed Data
Second Approach
Hope that the Central Limit Theorem applies.
This is a function of data skewness and the sample size
See Chapter 2 for Central Limit Theorem discussion

16. Bootstrapping
Currently the best way to compute a confidence internal from skewed data, or small sample size
Does not require assumption of normality

17. Confidence Intervals on Skewed Data
Third Approach - Bootstrapping
Sample from the data set, with replacement
The subsample is generated with replacement so that any data point can be sampled multiple times or not sampled at all.
Compute the estimated statistic
Do this many times
Confidence endpoints determined from the ranked estimated statistic
Based on the data set, so it works best with more data

18. Confidence Intervals on Skewed Data
Bootstrapping Example: Arsenic Data Set
Randomly pick 25 values from a 25 point arsenic data set. Sample with replacement.
Compute the mean of these 25 values
Do again 1000 times
A 2-sided 95% confidence interval for the mean is the 0.025*1000th and 0.975*1000th ranked values for the mean

19. Confidence Intervals on Skewed Data
Bootstrapping Example: Arsenic Data Set

20. Confidence Intervals on Skewed Data
Bootstrapping Example: Arsenic Data Set

21. Other Confidence Intervals
Can have other confidence intervals for other parameters
Variance
Standard Deviation
Other percentiles
Median
Confidence intervals for a percentile is often call a “tolerance interval”

22. Prediction Intervals
(contains one or more future observations with a specified probability)
Simplest prediction interval (nonparametric) is to use the percentiles of the data set
Two-sided 90% prediction interval  use the 5th and 95th percentiles
90% of the observed data fall within this interval, and thus we expect that 90% of the future observations will also fall within this interval
Requires ample data

23. Prediction Intervals
Parametric prediction interval will be shorter than a nonparametric interval if:
Data follow the distribution assumed by the interval calculation
Easy method for prediction interval
Transform data to look normal
Compute interval
Transform interval back to original units

24. Confidence vs. Prediction Intervals
A prediction interval will always be larger than the confidence interval for the same alpha.
Why? The mean of 10 observations, for example, is always less variable than the location of the 10 observations themselves.

25. Tolerance Intervals
(contains a proportion of future observations with a specified probability)
An interval around a proportion of the distribution
The proportion is called the “converge”
What cutoff(s) will cover 95% of all future observations, with 90% confidence?
Easy method for tolerance interval
Transform data to look normal
Compute interval
Transform interval back to original units
Works for prediction and tolerance intervals, but not confidence intervals

26. MINITAB Laboratory 2 Reading Assignment
Chapter 3 Describing Uncertainty (pages 65 to 96)
Statistical Methods in Water Resources
by D.R. Helsel and R.M. Hirsch
MINITAB
Laboratory 2