What is statistics? Why do engineers need statistics?

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1 ENGINEERING STATISTICS What is statistics? Why do engineers need statistics? Engineers build, design, operate, and/or improve physical systems and products. When theory fails, the engineer may need to collect and interpret data to help understand the process Statistics is the study of how best to: A. collect data; B. summarize or describe data; C. draw formal inferences and practical conclusions on the basis of data (all the while recognizing the reality of variation) As a preliminary step to a statistical analysis, one should: Identify the research objective - what is the question to be answered and the group of individuals that we want to make statements about, the group of interest or population. Then we proceed with the process: collect the information to answer the questions, summarize the information, and make conclusions. A. Collect the information needed to answer the questions - Gaining access to the entire population may pose problems, and thus we typically look at a subset of the population, called a sample, to observe the variable of interest Example: Want opinion on issue Variable of interest = opinion One observation = one student s opinion Sample = students giving opinion Population = who do we want to generalize to? Possibilities are: Everyone at University; Engineering students; Male students; Undergrads; ENGR 305 students Population could be conceptual: Example: Lifetime of a light bulb Variable of interest = lifetime (in hours) One observation = the lifetime of one light bulb Sample = 30 light bulbs (30 lifetimes) Population = all light bulbs that could be manufactured o In an Enumerative study, we have a finite population (e.g. population is our class) o In an Analytical study, we have an infinite/conceptual population (does not all exist in one time/ place)

2 Why do we take samples (instead of observing the whole population)? o The population may be too large o Time restrictions o The population might be conceptual like in the example above o Impractical (the experiment breaks or uses up what we are testing) o Limited resources to collect accurate data o Population might be inaccessible B. Organize and summarize the information One way is to give descriptive statistics that describe the data through numerical measurements, tables, charts, graphs. When collecting data we observe values of one or more variable(s). We want to know about the distribution of the variable(s); that is, the possible values and the corresponding prevalence of different (sets of) possible values. Sometimes we might settle for summaries of the distribution: Summaries of the distribution of the whole population are called parameters Summaries of the distribution of the sample (observed values) only are called statistics C. Draw conclusions from the information -- the information collected from the sample is generalized to the population and their reliability is measured, i.e. inferential statistics. Example: a researcher is conducting a study on average miles per gallon (highway) of a certain car produced at a factory (population: all cars of that type that can be produced), and obtains a sample of 100 cars of that type. The results obtained from the sample would be generalized to the population (which in this case is conceptual). The average miles per gallon (mpg) for the sample (a statistic) would be used to estimate the corresponding parameter (average mpg) for the population. There is always uncertainty when using samples to draw conclusions regarding a population because we can t learn everything about a population by looking at a sample. Therefore, statisticians will report a level of confidence in their conclusions. This level of confidence is a way of representing the reliability of results. In the context of estimation, what will be reported is an interval about the estimate (a confidence interval). If the claim is made by a consumer advocate that the car gets less than a certain number of miles per gallon, then one might want to test that hypothesis. There, the level of confidence would be manifested as a probability of making a wrong conclusion. Also, there are other topics that may arise in statistical analysis of data: Is one variable causing changes in another? Or, are variables highly correlated?

3 Some terminology: types of variables & data Qualitative or categorical variables allow for classification of individuals based on some attribute or characteristic Quantitative variables provide numerical measure Example: Determine whether the following are qualitative or quantitative Gender (qualitative), temperature (quantitative), number of days in the past week that a student went to class (quantitative), lifetime of a light bulb (quantitative) A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of values (can be lined up with 0,1,2,3, ). Counts of the number of occurrences of an event are a classic example of discrete variables A continuous variable is a quantitative variable that has an uncountably infinite number of possible values (the variable takes values in intervals, a continuum) The lifetime of a light bulb is continuous, though we tend to make it discrete by grouping (bins) into the number of days, etc. Types of data studies: 1) Observational study investigator s role is basically passive. Individuals in a sample are studied but no attempt is made to manipulate or influence the variables of interest. This type of study is good for establishing whether two variables are related, or to learn characteristics of a population. Observational studies are carried out when control is unethical or impossible. 2) Experimental study (Designed experiment) investigator s role is active. In a designed experiment, variables are manipulated, the study environment is regulated. Treatments are applied to experimental units, to try to determine the effects of the treatment on the response variable. This type of study is better for establishing causation. Example: To determine whether there is a connection between drinking and lung cancer, individuals are asked whether or not they smoke their rate of cancer is monitored. The individuals are not controlled in terms of their eating habits, how much they drink, etc. If there is a significant difference between drinkers and non-drinkers cancer rates, the researcher may claim that drinking causes cancer (actually they have determined that the two are associated). However, drinkers could have some characteristic (e.g. amount of exercise, diet, hanging out in smoky bars, lurking variables) that differs from the non- drinking group, and that is the cause of the cancer.

4 To do this as an experimental study, we would need to randomly divide the population into two groups, and, e.g. require one group to drink a certain amount each day for the next 20 years. We could then control for other factors that aren t under our control in an observational study, e.g. we could assign the same diet and exercise regimen, allow no smokers, etc. On the spectrum of studies, the experimental end is preferred as opposed to an observational study, but at times an observational study is the best we can do (for e.g., we wouldn t want to make people smoke or drink) In order to study the relationships among variables, observational studies are performed. Unlike controlled experimental designs where only certain variables are allowed to vary (at pre-specified levels), in observational studies the data on the variables are observed after the fact and recorded. Cause and effect are hard (and often impossible) to establish. But associations and predictabilities among variables can be investigated. Such associations and predictabilities may be further studied in a lab setting.

5 Sampling The goal in sampling is to obtain individuals in such a way that accurate information may be obtained about the population. Below we give basic terminology relative to the information that we obtain. Then we discuss a basic sampling technique that has certain good properties, and also a sampling technique that has properties that are not good. A measurement or measuring method is called valid if it usefully or appropriately represents the feature of an object or system that is of engineering importance. A measurement is called accurate (or unbiased) if on average it produces the true or correct value of a quantity being measured. A measurement system is called precise if it produces small variation in repeated measurement of the same subject.

6 Simple Random Sampling (SRS) o Every group of n distinct units of N in the population has an equal chance of being selected o As a consequence: every unit in the population has an equal chance of being selected to be in the sample Why: o Random sampling avoids selection bias. (An example of sampling bias is the following: suppose I am producing a drug and want to show that it has good effects. I can select the healthier or younger patients as the group to take my drug, making it appear that the drug gave positive effects.) o With simple random sampling you can quantify the general effects of sampling Note that simple random sampling does not guarantee a good or representative sample every time; we can get all small values or all large values. Sampling only guarantees certain long-run behavior of the estimates (on average the estimates will be unbiased). How: Paradigm: drawing names out of a hat, or using some randomized mechanism (in practice we could use computer generated random numbers) Example: Draw a sample of size 2 from a population of 5 students with mean weight 160 Data (weights) 100, 110, 150, 200, 240 The population mean is 160. (Note also the variation in weights for the population (variation is another population parameter) Taking the 10 samples of size 2, we compute x bar (the sample mean) for each sample: (1,2) x bar = 105 (1,3) x bar = 125 (1,4) x bar = 150 (1,5) x bar = 170 (2,3) x bar = 130 (2,4) x bar = 155

7 (2,5) x bar = 175 (3,4) x bar = 175 (3,5) x bar = 195 (4,5) x bar = 220 Notice the variation in x bar. Though the actual average weight is 160, if we take sample (1,2) (the first two people), the average weight is 105, but if we take sample (4,5), the average is 220. This is a property of simple random sampling that holds in general. Suppose I wanted information on an opinion from students in the class. Why select a SRS of 5 students to represent the class, instead of taking samples using other means? o There is bias in selecting students in front or back o There is bias in selecting students whose names I know o There also could be bias in using a sample of convenience Why Not SRS? o It is not always feasible, e.g. what if population is conceptual? o We need a frame (a list of all the elements to be sampled) Example: If we want to take a SRS of people who run at a particular track o How do we get a frame? o How do we randomize selection? Often we take a sample of convenience (individuals are easily obtained). The most popular convenience sample is one in which the individuals in the sample are self- selected, i.e. the individuals themselves decide to participate in the survey. These are also called voluntary response surveys. Examples include phone-in polling and Internet surveys. This is not a good sampling design, and thus we should be careful in generalizing the conclusions from them to the entire population.