MATH408: Probability & StatisticsSummer 1999

WEEK 2

Dr. Srinivas R. ChakravarthyProfessor of Mathematics and Statistics

Kettering University(GMI Engineering & Management Institute)

Flint, MI 48504-4898Phone: 810.762.7906

Email: schakrav@kettering.eduHomepage: www.kettering.edu/~schakrav

SAMPLE

• Sample: is a subset (part) of the population.

• Since it is infeasible (and impossible in many cases) to study the entire population, one has to rely on samples to make the study.

• Samples have to be as representative as possible in order to make valid conclusions about the populations under study.

SAMPLE (cont'd)

• Contain more or less the same type of information that the population has.

• For example if workers from three shifts are involved in assembling cars of a particular model, then the sample should contain units from all three.

• Samples will be used to “estimate” the parameters.

SAMPLE (cont’d)

• Much care should be devoted to the sampling.• There is always going to be some error involved in

making inferences about the populations based on the samples.

• The goal is to minimize this error as much as possible.

• There are many ways of bringing in systematic bias (consistently misrepresent the population).

SAMPLE (cont’d)

• This can be avoided by taking random samples.• Simple random sample: all units are equally likely to

be selected.

• Multi-stage sample: units are selected in several stages.

• Cluster sample: is used when there is no list of all the elements in the population and the elements are clustered in larger units.

SAMPLE (cont’d)

• Stratified sample: In cases where population under study may be viewed as comprising different groups (stratas) and where elements in each group are more or less homogeneous, we randomly select elements from every one of the strata.

• Convenience sample: samples are taken based on convenience of the experimenter.

• Systematic sample: units are taken in a systematic way such as selecting every 10th item after selecting the first item at random.

HOW TO USE SAMPLES?

• Samples should represent the population.

• Random sample obtained will not always be an exact copy of the population.

• Thus, there is bound to be some error:

SAMPLES (cont’d)

• Random or unbiased error: This is due to the random selection of the sample and the mean of such error will be 0 as positive deviation and negative deviation cancel out. This random error is also referred to as random deviation and is measured by the standard deviation of the estimator.

• Non-random or biased error: this occurs due to several sources such as human, machines, mistakes due to copying or punching, recording and so on. Through careful planning we should try to avoid or minimize this error.

EXPERIMENTS USING MINITAB

• We will illustrate the concepts of sample, sampling error, etc with practical data using MINITAB when we go to the laboratory next time.

NEXT?

• Once the data has been gathered, what do we do next?

• Before any formal statistical inference through estimation or test of hypotheses is conducted, EDA should be employed.

EXPLORATORY DATA ANALYSIS (EDA)

• This is a procedure by which the data is carefully looked for patterns, if any, and to isolate them.

• First step in identifying appropriate model.

EDA(cont'd)

• The main difference between EDAand conventional data analysis is:

– while the former, which is more flexible (in terms of any assumptions on the nature of the populations from which the data are gathered) emphasizes on searching for evidence and clues for the patterns, the latter concentrates on evaluating the evidence and the hypotheses on the nature of the parameters of the population(s) under study.

CAPABILITY ANALYSIS

• Deals with the study of the ability of the process to manufacture products within specifications.

• In order to perform the capability analysis, the process must be stable (i.e., things such as warm up period needed on the process before manufacturing products and others should be taken care of).

CAPABILITY ANALYSIS (cont'd)

• The process specifications are compared to the variance (or the spread) of the process.

• For a process to be more capable, more measurements would be expected to fall within the specifications.

CAPABILITY ANALYSIS (cont'd)

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A commonly used capability index is given by

CAPABILITY ANALYSIS (cont'd)

• The larger the value of Cpk, the less evidence that the process is outside the specifications. A value of 1.5 or higher for Cpk is usually desired. More on this will be seen later.

DESCRIPTIVE STATISTICS

• Deals with characterization and summary of key observations from the data.

• Quantitative measures: mean, median, mode, standard deviation, percentiles, etc.

• Graphs: histogram, Box plot, scatter plot, Pareto diagram, stem-and-leaf plot, etc.

• Here one has to be careful in interpreting the numbers. Usually more than one descriptive measure will be used to assess the problem on hand.

DIFFERENT TYPES OF PLOTS

• Point plot: The horizontal axis (x-axis) covering the range of the data values and vertically plot the points, stacking any repeated values.

• Time series plot: x-axis corresponds to the number of the observation or the time of the observation or the day and so on and the y-axis will correspond to the value of the observation.

Time-series plot

PLOTS (cont'd)

• Scatter plot: Construct x-axis and y-axis that cover the ranges of two variables. Plot (x i, yi) points for each observation in the data set.

• Histogrom: This is a bar graph, where the data is grouped into many classes. The x-axis corresponds to the classes and the y-axis gives the frequency of the observations.

Histogram

PLOTS (cont'd)

• Stem-and-leaf plot: Data is plotted in such a way the output will look like histogram and also features a frequency distribution. The idea is to use the digits of the data to illustrate its range, shape and density. Each observation is split into leading digits and trailing digits. All the leading digits are sorted and listed to the left of a vertical line. The trailing digits are written to the right of the vertical line.

Stem-and-leaf

PLOTS (cont'd)

• Pareto Diagram: Named after the Italian economist. This is a bar diagram for qualitative factors. This is very useful to identify and separate the commonly occurring factors from the less important ones. Visually it conveys the information very easily.

Pareto Diagram

40 56 64 80120440 5.0 7.0 8.010.015.055.0

100.0 95.0 88.0 80.0 70.0 55.0

100

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400

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Pareto Chart for : FailuresSolder Defects (Messina:SQC for MM, 1987)

PLOTS (cont'd)

• Box plot: is due to J. Tukey and provides a great deal of information. A rectangle whose lower and upper limits are the first and third quartiles, respectively, is drawn. The median is given by a horizontal line segment inside the rectangle box. The average value is marked by a symbol such as “x” or “+”. All points that are more extreme are identified.

Boxplot for MPG example

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Box Plot of City MPG vs Size

PLOTS (cont'd)

• Quantile plot: This plot is very useful when we want to identify/ verify an hypothesized population distribution from which the data set could have been chosen. A quantile, Q(r), is a number that divides a sample (or population) into two groups so that the specified fraction r of the data values is less than or equal to the value of the quantile.

PLOTS (cont'd)

• Probability plot: This involves plotting the cumulative probability and the observed value of the variable against a suitable probability scale which will result in linearization of the data. The basic steps involved here are: (a) Sorting the data into ascending order; (b) Computing the plotting points; (c) Selecting appropriate probability paper; (d) Plot the points; (e) Fitting a “best” line to data.

MEASURES OF LOCATION

• MEAN: Used very often in analyzing the data.

– Although this is a common measure, if the data vary greatly the average may take a non-typical value and could be misleading.

• Median: is the halfway point of the data and tells us something about the location of the distribution of the data.

• Mode: if exists, gives the data point that occur most frequently.

– It is possible for a set of data to have 0, 1 or more modes.

LOCATION (cont’d)

• Mean and median always exist.• Mode need not exist.• Median and mode are less sensitive to extreme

observations.• Mean is most widely used.• There are some data set for which median or mode

may be more appropriate than mean

LOCATION (cont’d)

• Percentiles: The 100pth percentile of a set of data is the value below which a proportion p of the data points will lie.

• Percentiles convey more information and are very useful in setting up warranty or guarantee periods for manufactured items.

• Also referred to as quantiles.• The shape of the frequency data can be classified

into several classes.

LOCATION (cont’d)

• Symmetric: mean = median = mode• Positively skewed: tail to the right; mean > median• Negatively skewed:tail to the right; median >

mean• In problems, such as waiting time problems one is

interested in the tails of the distributions.• For skewed data median is preferred to the mean.

MEASURES OF SPREAD

• One should not solely rely on mean or median or mode.

• Also two or more sets of data may have the same mean but they may be qualitatively different.

• In order to make a meaningful study, we need to rely on other measures.

MEASURES OF SPREAD

• For example, we may be interested to see how the data is spread.

• Range: is the difference between the largest and the smallest observations.

• Quick estimate on the standard deviation.

• Plays an important role in SPC.

SPREAD (cont’d)

• Standard deviation: describes how the data is spread around its mean.

• Coefficient of variation: The measures we have seen so far depend on the unit of measurements. It is sometimes necessary and convenient to have a measure that is independent of the unit and such a useful and common measure is given by the ratio of the standard deviation to the mean called the coefficient of variation.

SPREAD (cont’d)

• Interquartile range: is the difference between the 75th and 25th percentiles.

• Gives the interval which contains the central 50 % of the observations.

• Avoids the total dependence on extreme data

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Stem-and-leaf of cycles N = 70Leaf Unit = 100 (Problem 2.2)

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400 800 1200 1600 2000

95% Confidence Interval for Mu

1250 1350 1450 1550

95% Confidence Interval for Median

Variable: cycles

A-Squared:P-Value:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

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Anderson-Darling Normality Test

95% Confidence Interval for Mu

95% Confidence Interval for Sigma

95% Confidence Interval for Median

Descriptive StatisticsProblem 2.2

INFERENTIAL STATISTICS

• Recall that a parameter is a descriptive measure of some characteristic of the population.

• The standard ones are the mean, variance and proportion.

• We will simply denote by , the parameter of the population under study.

INFERENTIAL STATISTICS

• Estimation Theory and Tests of Hypotheses are two pillars of statistical inference.

• While estimation theory is concerned about giving point and interval estimates for parameter(s) under study, test of hypotheses deals with testing claims on the parameter(s).

Illustrative Example 1

• The following data corresponds to an experiment in which the effect of engine RPM on the horsepower is under study.

TABLE 1: Data for HP Example

hp@4500 243 241 233 264 233 263 248 240hp@5500 211 234 218 240 218 237 216 203hp@4500 258 257 247 276 248 272 261 268hp@5500 252 264 245 275 244 270 257 273

Variable N Mean Median Tr Mean StDev SE Mean

hp@4500 16 253.25 252.50 253.07 13.51 3.38hp@5500 16 241.06 242.00 241.36 23.16 5.79

Variable Min Max Q1 Q3

hp@4500 233.00 276.00 241.50 263.75hp@5500 203.00 275.00 218.00 262.25

302010

280

270

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Time Order

Hor

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Horse Power at 4500 RPM and 5500 RPM

hp@5500hp@4500

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Boxplots of hp@4500 and hp@5500(means are indicated by solid circles)

GROUNDWORK FOR PROBABILITY

• Looking at the data in Table 1, why is that the hp values, say at 4500 RPM, are not exactly the same if the experiment is repeated under the “same conditions”?

• The fluctuation that occurs from one repetition to another is called experimental variation, which is usually referred to as “noise” or “statistical error” or simply “error” [Recall this term from earlier discussion on data collection].

PROBABILITY (cont’d)

• This represents the “variation” that is inherently present in any (practical) system.

• The noise is a “random variable” and is studied through probability.

What is Probability?

• A manufacturer of blender motors wants to determine the warranty period for this product.

• If motor life were constant, (say 8 years) the manufacturer would have no problem. The motor could be warranted for 8 years.

• But, in reality, the motor life is not a constant.

PROBABILITY (cont’d)

• Some motors will fail quickly and others will last for several years.

• There is an element of randomness in the life of the motors.

• The manufacturer cannot precisely predict how long any motor will last.

• Probability theory gives the manufacturer the means to quantify what is known about motor lifetimes and helps to quantify the risks involved in setting a warranty period.

PROBABILITY (cont’d)

• Similar problems arise in the context of other products.

• FMS play an important role in modern manufacturing. Improved quality, lower inventory, shorter lead times, higher productivity and greater safety are some of the benefits derived from FMS.

• All of these have random elements.

PROBABILITY (cont’d)

• Probability theory deals with randomness, allowing the study of quantities whose behavior cannot be predicted completely in advance.

• The above examples deal with manufacturing.

PROBABILITY (cont’d)

• We could just as easily find examples in business, electrical and computer engineering, biomedical science and engineering, sociology, economics, marketing, civil engineering, the behavioral sciences and so on. The underlying problem, randomness, is the same.

PROBABILITY (cont’d)

• One should understand the ideas of probability and statistics from both theoretical and practical points of view.

• To properly apply probability and statistics in the real world, we must appreciate both sides of the picture.

• We cannot properly apply a procedure if we don't, at least in general terms, understand the reasoning (theory) behind it.

PROBABILITY (cont’d)

• On the other hand, trying to apply theory without knowledge of the area of application is foolish. We have to have a proper perspective on both before meaningful progress can be made.

• Probability theory develops mathematical models for random experiments.

• A random experiment is a sequence of actions whose outcome cannot be predicted with certainty.

PROBABILITY (cont’d)

• If you've used phrases like "one chance in a 1000", "50-50" or "3-to-2 odds" to describe something, you have most likely been using an informal probability model.

• If we throw two fair dice and our concern is about whether or not the dice eventually land and come to rest, then the throwing of the two dice is not a random experiment.

• Our knowledge of physical laws allows us to predict with

virtual certainty that this outcome will happen.

PROBABILITY (cont’d)

• If, however, we are concerned with how many dots show on the topmost faces when the dice come to rest, then we are performing a random experiment in tossing the dice, since we cannot predict with certainty which faces will show.

PROBABILITY (cont’d)

• Outcomes of random experiments: the length of a phone call, the gender mix of three people chosen from a group of 25 people, and the phenotype of the offspring of a cross breeding experiment, the number of defects on a painted panel.

EXPERIMENT

• Calculation of MPG of a new model car.• Measurements of current in a thin copper wire.• Measurements of Film build thickness in a

painting process.• Duration of phone calls.• Time to assemble a job.• Tossing a coin.• Throwing a dice.

Sample space (S)

• Collection of all possible outcomes in an experiment.– The MPG’s of all cars from that particular

model car.

• Event (A)– A subset of a sample space

– The MPG of the new model car exceeds, say, 25 miles.

SET THEORY

AB AB

SET THEORY (cont’d)

A'

PROBABILITY

• is a function defined on the set of all possible events.

• is a number between 0 and 1.

• satisfies a set of axioms:– P(A) 0.– P(S) = 1.

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PROPERTIES

• 0 P(A) 1.

• P(A') = 1 - P(A)

• P(AB) = P(A) + P(B) - P(AB).

Classical definition

• While axiomatic definiton of probability is very useful in developing the theory of probability, it doesn’t tell us how to compute probabilities of events.

• Classical Definition: If S has a finite number of sample points and are equally likely to occur, then P(A) = number of points in A / number in S.

• If S doesn’t contain equally likely outcomes, then P(A) = sum of the weights associated with points in A.

Classical definition(cont’d)

• To use this definition, we need to calculate the number of points in S and in A.

• How do we do this without actually listing all possible outcomes?

• Using Counting Techniques.– Principle of addition and multiplication

– Permutations and combinations

Principle of Addition and Multiplication

• If the task is done if any one of the subtasks is done, then the total number of ways of doing the main task is n1 + n2 + ... + nk .

• If the task is done if and only if all the subtasks are done, then the total number of ways of doing the main task is the product n1 n2 ... nk.

PERMUTATION

• Suppose that r objects are to be drawn without replacement from n (r n).

• If the order of selection is important, then using the principle of multiplication we see that the total number of ways of doing this is n(n-1)...(n-r+1).

• This could be written in a compact form using the factorials as n!/(n-r)! or Pr

n.

COMBINATION

• Suppose that r objects are to be drawn without replacement from n (r n).

• When the order of selection is not important, any particular set of r objects can be ordered in Pr

r = r! ways, the total number of ways of selecting r out of n in which order is immaterial is Pr

n /r!. It is convenient to denote this by Cr

n or by

r

n

EXAMPLES

CONDITIONAL PROBABILITY

• What we saw so far is referred to as unconditional probability. That is, the probabilities of events of interest were computed only based on the sample space and with no prior information.

• Sometimes it is convenient to compute certain unconditional probabilities by first conditioning on some event.

CONDITIONAL PROBABILITY

• Also, this plays an important role in stochastic modeling.

• In a finite buffer queuing model, computation of waiting time of an admitted customer involves conditional probability.

• DEFINITION: P(B/A) = P(AB) / P(A)

• Events A and B are independent if and only if P(AB) = P(A)P(B).

RANDOM VARIABLES

• Often in probability and statistics, the quantities that are of interest are not the outcomes but rather the values associated with the outcome of the experiment.

• If n items are selected from a production lot the quality inspector is interested in the number of defectives out of the n chosen and the corresponding probabilities.

RANDOM VARIABLES (cont’d)

• A random variable, X, is a real-valued function defined on the sample space S into the set of real numbers.

• Random variables can be – Discrete – taking only discrete values

• Number of defective molds

– Continuous – taking continuous values• Time taken to assemble a product• Mixture of discrete and continuous• Waiting time of a customer

INDEPENDENCE

EXAMPLES

STUDY OF RANDOM VARIABLES

• Probability functions

• Probability mass function (discrete)

• Probability density function (continuous)

• Cumulative probability distribution function