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Understanding Data Analysis Essentials: Hypothesis Testing, Confidence Intervals & Paired Samples

Delve into the core concepts of data analysis, including hypothesis testing, confidence intervals, and paired samples. Explore how to quantify population properties statistically, apply the Central Limit Theorem, and conduct hypothesis tests for population means with both large and small sample sizes.

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Understanding Data Analysis Essentials: Hypothesis Testing, Confidence Intervals & Paired Samples

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  1. Data Analysis

  2. Basic Problem • There is a population whose properties we are interested in and wish to quantify statistically: mean, standard deviation, distribution, etc. • The Question – Given a sample, what was the random system that generated its statistics?

  3. Central Limit Theorem • If one takes random samples of size n from a population of mean m and standard deviation s, then as n gets large, approaches the normal distribution with mean m and standard deviation • s is generally unknown and often replaced by the sample standard deviation s resulting in , which is termed the Standard Error of the sample.

  4. Example

  5. Normal distribution

  6. Critical Values for Confidence Levels

  7. Confidence Interval for Mean(large sample size) OR

  8. Student’s t-distribution

  9. Critical Values for Confidence Levelst-distribution

  10. Confidence Interval for Mean(small sample size, t-distribution) OR

  11. Comparing Population Means Unequal Variance Pooled Variance

  12. Hypothesis Testing (t-test) • Null Hypothesis – differences in two samples occurred purely by chance • t statistic = (estimated difference)/SE • Test returns a “p” value that represents the likelihood that two samples were derived from populations with the same distributions • Samples may be either independent or paired

  13. Tails • One tailed test – hypothesis is that one sample is: less than, greater than, taller than, • Two tailed test – hypothesis is that one sample is different (either higher or lower) than the other

  14. Paired Test • Samples are not independent • Much more robust test to determine differences since all other variables are controlled • Analysis is performed on the differences of the paired values • Equivalent to Confidence interval for the mean

  15. Paired Samples – New Site

  16. TSS Concentrations vs. Time

  17. BMP Performance Comparison • Commonly expressed as a % reduction in concentration or load • Highly dependent on influent concentration • Potentially ignores reduction in volume (load) • May lead to very large differences in pollutant reduction estimates • Preferable to compare discharge concentrations

  18. Effect of TSS Influent Concentration

  19. Sand Filter - TSS

  20. Comparison of Effluent Quality

  21. Exercise • Calculate average concentrations for each constituent for the two watersheds • Determine whether any concentrations are significantly different, report p value for null hypothesis • Calculate average effluent concentrations for the two BMPs and determine whether they are different from the influent concentrations – p values • Compare effluent concentrations for the two BMPs and determine whether one BMP is better than the other for a particular constituent.

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