Statistical Inference “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” (H.G. Wells, 1946) “There are three kinds of lies: white lies, which are justifiable; common lies, which have no justification; and statistics.” (Benjamin Disraeli) “Statistics is no substitute for good judgment.” (unknown)

Statistical Inference Suppose – A mechanical engineer is considering the use of a new composite material in the design of a vehicle suspension system and needs to know how the material will react under a variety of conditions (heat, cold, vibration, etc.) An electrical engineer has designed a radar navigation system to be used in high performance aircraft and needs to be able to validate performance in flight. An industrial engineer needs to validate the effect of a new roofing product on installation speed. A motorist must decide whether to drive through a long stretch of flooded road after being assured that the average depth is only 6 inches.

Statistical Inference What do all of these situations have in common? How can we address the uncertainty involved in decision making? a priori a posteriori uncertainty (due to variability in the system, in the measurement, etc.) a priori – based on understanding the characteristics of the system, ‘expert opinion’, etc. posteriori – based on frequencies, experimentation, data

Probability A mathematical means of determining how likely an event is to occur. Classical (a priori): Given N equally likely outcomes, the probability of an event A is given by, where n is the number of different ways A can occur. Empirical (a posteriori): If an experiment is repeated M times and the event A occurs mA times, then the probability of event A is defined as, P(A) = n/N P(A) = mA / M

The Role of Probability in Statistics In statistical inference, we want to make general statements about the population based on measurements taken from a sample. How will all suspension systems produced with the new composite behave? How will the radar navigation system perform in all aircraft? What speed improvements will we obtain for all roofing applications using the new product? To answer these questions, we ___________ from the population and hope to generalize the results.

Observations & Statistical Inference Example, An experiment is designed to determine how long it takes to install a roof using a new product. Experiment Design Result: t = 2.32 sec/ft2, P = 0.023 p – value: experiment = a prescribed set of observations in which critical factors (independent variables) are manipulated in order to determine the effect on the output (dependent variable) Design = specification of hypotheses, factors, factor levels, and sample size. Related terms – randomized design, random sampling, variability p-value = the probability that the results obtained could have happened by chance (that is, that this is not a ‘real’ result) small p indicates significant result (that is, we’re pretty sure this is ‘real’) small generally means <.05 (but this is subject to experimenter judgment)

Descriptive Statistics Numerical values that help to characterize the nature of data for the experimenter. Example: The absolute error in the readings from a radar navigation system was measured with the following results: the sample mean, x = _________________________ the sample median, x = _____________ 17 22 39 31 28 52 147 ~