Ralph Rackstraw, your next-door neighbor, surprised his wife last January by having a hot tub installed in their back yard while she was away on a business trip. It surprised her, all right, but instead of being pleased she was horrified. "Have you lost your mind, Ralph?" she sputtered. "It will cost a fortune to keep this thing hot." "Don't be silly, Josephine," he replied. "It can't cost more than pennies a day, even in the dead of winter." "No way-and when did you become such an expert, anyway?" "I guarantee it will cost nothing-and I don't see your Ph.D. certificate on the kitchen wall either." They argued for awhile and then. remembering your chemical engineering education, came to ask you to seUle it for them. You asked a few questions, made several observations, converted everything to metric units, and arrived at the following data, all corresponding to an average outside air temperature of about 5C. The tub holds 1230 liters of water. Rackstraw normally keeps the tub temperature at 29C, raises it to 40C when he plans to use it, keeps it at 40C for about one hour, and drops it back to 29C when he is finished. During heating, it takes about three hours for the water temperature to rise from 29C to 40C. When the heat is shut off, it takes eight hours for the water temperature to drop back to 29C. Electricity costs 10 cents per kilowatt-hour. Taking the heat capacity of the tub contents to be that of pure liquid water and neglecting evaporation, answer the following questions. (a) What is the average rate of heat loss (kW) from the tub to the outside air? (Hint: Consider the period when the tub temperature is dropping from 40C to 29C.) (b) At what average rate (kW) does the tub heater deliver energy to the water when raising the water temperature? What is the total quantity of electricity (kWh) that the heater must deliver during this period? [Consider the result of part (a) when performing the calculation.] (c) (These answers should settle the argument.) Consider a day in which the tub is used once. Use the results of parts (a) and (b) to estimate the cost ($) of heating the tub from 29C to 40C and the cost ($) of keeping the tub at a constant temperature. (There is no cost for the period in which T is dropping.) What is the total daily cost of running the tub? Assume the rate of heat loss is independent of the tub temperature. (d) The tub lid, which is an insulator, is removed when the tub is in use. Explain how this fact would probably affect your cost estimates in part (c)
Lecture 1: Statistics Info and some Basic Principles -‐ Statistics is the most important science in the whole world: for upon it depends the practical application of every other science and of every art: the one science essential to all political and social administration, all education, all organization based on experience, for it only gives results of our experience." Florence Nightingale, Statistician -‐ Statistics are numbers measured for some purpose. -‐ Statistics is a collection of procedures and principles for gathering data and analyzing information in order to help people make decisions when faced with uncertainty. -‐ Course Goal: Learn various tools for using data to gain understanding and make sound decisions about the world around us. -‐ Chapter 1 starts out with eight statistical stories with morals, presented as seven case studies. -‐ In each, data are used to make a decision, a judgment, about a situation. These case studies follow a wide range of ideas and methods and introduce a lot of statistical language. -‐ 1. Who are those speedy drivers Principle: Simple summaries of data can tell an interesting story and are easier to digest than long lists. 2. Safety in the Skies Principle: When discussing the change in the rate or risk of occurrence of something, make sure you always include baseline or base rates. 3. Did anyone ask whom you’ve been dating Principle: A representative sample of only a few thousand, or perhaps even a few hundred, can give reasonably accurate information about a population of many millions. 4. Who are those angry women Principle: An unrepresentative sample, even a large one, tells you almost nothing about the population. 5. Does prayer lower blood pressure Principle: Cause-‐and-‐effect conclusions cannot generally be made on the basis of an observational study. 6. Does Aspirins reduce heart attack rates Principle: Unlike with observational studies, cause-‐and effect conclusions can generally be made on the basis of randomized experiments. 7. Does the internet increase loneliness and depression Principle: A statistically significant finding does not necessarily have practical significance or importance. When a study reports a statistically significant finding, find out the magnitude of the relationship or difference. 8. Did your mother’s breakfast determine your sex Principle: For studies that found a relationship or difference, find out how many different things were tested. The more tests done, the more likely a statistically significant difference is a false positive that can be explained by chance. Watch out if many things are tested and only 1-‐2 of them are statistically significant.