Lecture 3 Confidence Intervals and Experimental Objectives

1. Lecture 3Confidence IntervalsandExperimental Objectives

2. Confusion… • If I measured , can I say that I am 68% confident that the true mean, , lies within 1 standard deviation of ?

3. Confusion… • If I measured , can I say that I am 68% confident that the true mean, , lies within 1 standard deviation of ? • NO!!! • I can only say that 68% of the measurements lie within 1 standard deviation of (if they are normally distributed).

4. Confidence Intervals • The confidence interval is a range of possible values of the population mean, , calculated from the sample mean, , and standard deviation, . • Notice that the confidence interval uses sample statistics ( and ) to say something about the population mean ().

5. Calculation Lower bound Upper bound That second term looks like the standard error… was I overly critical of the standard error in my previous lecture???

6. Yes, I was Let me be a little more precise and less parochial…thanks to wikipedia. • standard error is an estimate of how close to the population mean your sample mean is likely to be – it behaves like a confidence interval. • standard deviation is the degree to which individuals within the sample differ from the sample mean (“Spread”). • Standard error should decrease with larger sample sizes, as the estimate of the population mean improves. Standard deviation will be unaffected by sample size.

7. Calculation Lower bound Upper bound Obtain ‘t’ from a table or, more likely, just let Excel do the work for you.

8. Excel • Tools -> Data Analysis -> Descriptive Statistics

9. Output • ‘Confidence Level’ on line 31 is • So, we are 95% confident that the true mean lies in

10. Question • If we ask Excel for a 90% confidence level, will the range be larger or smaller?

11. Question • If we ask Excel for a 90% confidence level, will the range be larger or smaller? • SMALLER – we are less confident so the true value is more likely to be outside the confidence interval.

12. Question • If we ask Excel for a 99% confidence level, will the range be larger or smaller? • Larger . The ‘Confidence Level (99%)’ is 0.342.

13. Question • How can we reduce the size of the confidence interval?

14. Question • How can we reduce the size of the confidence interval? • Smaller standard deviation or more samples.

15. Part 2: Experimental Objectives • A very clear, concise statement of the experimental objective, or hypothesis to be tested. • You may want to write an Introduction that includes both the motivation for the study and the experimental objective(s). • I like to italicize the experimental objective in reports so that it stands out from the rest of the introduction.

16. Types of Objectives • Principles of science can be tested • Theory can be demonstrated • Coefficients can be determined • Hypotheses can be tested • Assumptions can be validated • Effectiveness (or ineffectiveness) of the equipment for a particular task can be determined

17. Example #1 • The object of this experiment was to find the operating limits of a steam chamber with regards to the condensation of a steam to liquid water. A two-factor factorial design was used to find the operating limits. • Clear? • Concise?

18. Example #2 • The experimental objective was to find the temperature and volume operating limits for the Steam Chamber simulation. A two-factor, temperature and volume (T, V), factorial design was defined to find the operating limits. Four different values for T and V were chosen, giving sixteen possible combinations of temperature and pressure. Three replicates were also measured, for a total of 19 tests, to determine if the simulation is repeatable.

19. Example #3 • Perform an preliminary study on the effects of Volume and Temperature on condensation of steam.

20. Example #4 • The objective of this experiment was to find the temperature and volume operating limits for a Steam Chamber simulation. For this simulation, the temperature and volume were adjustable, then the pressure was measured inside the chamber at a given volume and temperature. A check on the vapor pressure allowed a warning to pop up when the vapor condensed at low enough temperatures and volumes.

21. Example #5 • A steam chamber simulation was tested using two factors, temperature and volume. The experiment will consist of four levels of volume, five levels of temperature, and three duplicates. These values will be 1 L, 3 L, 5 L, 6 L, 250°F, 310°F, 350°F, 370°F, and 470°F with duplicates at 1 L/250°F, 3 L/310°F, and 5 L, 350°F. By testing these points in the steam chamber simulation it can be determined for what values of volume and temperature the simulation is valid. This knowledge of were the simulation is valid is important, since at low volumes and temperatures the vapor pressure is exceeded and the fluid in the simulation condenses and therefore at certain ranges the simulation becomes invalid for calculations.

22. What does a report look like? • Abstract – more later • Introduction • Experimental Objective • Background (around objective) • Theory • Methods (many parts to this section) • Results/Analysis • Conclusions