Outline Why Statistics? Populations, Samples, and Census Proportions, Averages, Variances, and Percentiles Week 1 Basic Statistical Concepts, Part I Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Populations, Samples, and Census Proportions, Averages, Variances, and Percentiles Week 1 Objectives We will give an introduction to the statistical package R, and to statistics. The introduction to R is included in a different pdf file and a script file. After motivating the need for statistics in engineering and scientific research, we will introduce fundamental notions that forms the foundation for the material in later weeks. In particular we will introduce the notions of population, census, sampling, and sampling variability, and will define such basic quantities as the proportion, mean, variance and percentiles. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Populations, Samples, and Census Proportions, Averages, Variances, and Percentiles 1 Why Statistics? 2 Populations, Samples, and Census Some Sampling Concepts Random Variables and Statistical Populations 3 Proportions, Averages, Variances, and Percentiles Population Proportions and Sample Proportions Population Averages and Sample Averages Population Variance and Sample Variance Sample Percentiles Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Populations, Samples, and Census Proportions, Averages, Variances, and Percentiles Example (Examples of Engineering/Scientific Studies) Comparing the compressive strength of two or more cement mixtures. Comparing the effectiveness of three cleaning products in removing four different types of stains. Predicting failure time on the basis of stress applied. Assessing the effectiveness of a new traffic regulatory measure in reducing the weekly rate of accidents. Testing a manufacturer’s claim regarding a product’s quality. Studying the relation between salary increases and employee productivity in a large corporation. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Populations, Samples, and Census Proportions, Averages, Variances, and Percentiles These studies require Statistics due to the intrinsic variability: The compressive strength of different preparations of the same cement mixture will differ. The figure in http://personal.psu.edu/acq/401/fig/ HistComprStrCement.pdf shows 32 compressive strength measurements (MegaPascal units), of test cylinders (6 in. diameter, 12 in. high), using water/cement ratio of 0.4, measured on the 28th day after they are made. Under the same stress, two beams fail at different times. The proportion of defective items of a certain product will differ from batch to batch. Intrinsic variability renders the objectives of the case studies, as stated, ambiguous. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Populations, Samples, and Census Proportions, Averages, Variances, and Percentiles The objectives of the case studies can be made precise if stated in terms of averages or means. Comparing the average compressive strength of two different cement mixtures. Estimation of average failure time on the basis of stress applied. Estimation of the average proportion of defective items. Moreover, because of variability, the words ”average” and ”mean” have a technical meaning which can be made clear through the concepts of population and sample. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Definition Population is a well-defined collection of objects or subjects, of relevance to a particular study, which are exposed to the same treatment or method. Population members are called units. The objective of a study is to investigate certain characteristic(s) of the units of the population(s) of interest. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Example (Populations and Unit Characteristics) All water samples taken from a lake. Characteristics: Mercury concentration; Concentration of other pollutants. All items of a certain manufactured product (that have, or will be produced). Characteristic: Proportion of defective items. All students enrolled in Big Ten universities during the 2019-2020 academic year. Characteristics: Favorite type of music; Political affiliation. Two types of cleaning products. Characteristic: cleaning effectiveness. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Populations consisting of the same type of units but differ in the treatment, or method, applied to them are called treatment populations. Example (Treatment Populations) The concentration of pollutants in water samples is analyzed by two different labs. Water samples sent to Lab 1 constitute population 1, and those sent to Lab 2 constitute population 2. The time to failure of beams is studied under different stress conditions. The beams subjected to each stress condition constitute different populations. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Census and Samples Full (i.e., population-level) understanding of a characteristic can only be achieved by examining all population units. This is called census. However, taking a census can be time consuming and expensive: The 2000 U.S. Census costed $6.5 billion, while the 2010 Census costed $13 billion. Moreover, census is not feasible if the population is hypothetical or conceptual, i.e., not all members are available for examination. Because of the above, we typically settle for examining all units in a sample, which is a subset of the population. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Due to the intrinsic variability, the sample properties/attributes of the characteristic of interest will differ from the population-level properties/attributes. For example: The average mercury concentration in 25 water samples will differ from the overall mercury concentration in the lake. The proportion in a sample of 100 PSU students who favor expanding the use of solar energy will differ from the corresponding proportion of all PSU students. The relation between bear’s chest girth and weight in a sample of 10 bears, will differ from the corresponding relation in the entire population of 50 bears in a forested region. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles The GOOD NEWS is that, if the sample is suitably drawn, then sample properties approximate the population properties. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles 400 300 Weight 200 100 20 25 30 35 40 45 50 55 Chest Girth Figure: Population and sample relationships between chest girth and weight of black bears. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Sampling Variability Sample properties of the characteristic of interest also differ from sample to sample. For example: 1 The number of US citizens, in a sample of size 20, who favor expanding solar energy, will (most likely) be different from the corresponding number in a different sample of 20 US citizens. 2 The average mercury concentration in two sets of 25 water samples drawn from a lake will differ. The term sampling variability is used to describe such differences in the properties of the characteristic of interest from sample to sample. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles 400 300 Weight 200 100 20 25 30 35 40 45 50 55 Chest Girth Figure: Illustration of Sampling Variability. Week 1 Basic Statistical Concepts, Part I Outline Why Statistics? Some Sampling Concepts Populations, Samples, and Census Random Variables and Statistical Populations Proportions, Averages, Variances, and Percentiles Parameters and Statistics Population level properties/attributes of characteristic(s) of interest are called (population) parameters. Examples of parameters include averages, proportions, percentiles, and the correlation coefficient. The corresponding sample properties/attributes of characteristics are called statistics. Sample statistics approximate the corresponding population parameters but are not equal to them. Statistical inference deals with the uncertainty issues which arise in approximating parameters by statistics. The tools of statistical inference include point and interval