Statistics for Decision Making Normal Distribution: Reinforcement & Applications Instructor: John Seydel, Ph.D. QM Fall 2003

Student Objectives Discuss the characteristics of normally distributed random variables Calculate probabilities for normal random variables Apply normal distribution concepts to practical problems Recognize other common probability distributions Summarize the concept of sampling distributions

Administrative Items Grading: haven’t gotten very far Homework for today: not being collected Next week: Exam 2 Now, a quiz...

Quiz #1 Put your name in the upper right corner of the quiz Answer Problems 1 & 2 on the back side of the quiz You may refer To your homework But not to your text, notes, neighbor,... Normal probability table: on screen (ask if you need it adjusted)

Normal Distribution Review Description of the distribution Basic conceptsconcepts Determining probabilities Don’t forget the sketch

Some Quick Exercises: Mechanics Let x ~ N(34,3) as with the mpg problem Determine Tail probabilities  F(30) which is the same as P(x ≤ 30)  P(x > 40) Tail complements  P(x > 30)  P(x < 40) Other  P(32 < x < 33)  P(30 < x < 35)  P(20 < x < 30)

Questions About the Homework? Data analysis (Web Analytics case) Univariate Bivariate  Quantitative variables  Qualitative variables Normal distribution problems Basic mechanics Applications (e.g., Handout) Understanding the concepts Midterm exam from spring

Other Distributions Not everything is normally distributed Consider data in claimdat.xls Uniform distribution Exponential distribution Normal distribution Should be able to recognize shapes Also: be familiar with basic characteristics

Keep In Mind Probability = proportion of area under the normal curve What we get when we use tables is always the area between the mean and z standard deviations from the mean Because of symmetry P(x >  ) = P(x <  ) = Tables show probabilities rounded to 4 decimal places If z < then probability ≈ If z > 3.89 then probability ≈ Theoretically, P(x = a) = 0 P(30 ≤ x ≤ 35) = P(30 < x < 35)

Now, An Application to SamplingSampling Descriptive numerical measures calculated from the entire population are called parameters. Numeric data:  and  Categorical data: p (proportion) Corresponding measures for a sample are called statistics. Numeric data: x-bar and s Categorical data: p-hat

A Demonstration Draw a sample of 50 observations x ~ N(100,20) Calculate the average Note that x-bar doesn’t equal  Repeat multiple times Average the averages Look at the distribution of the averages Take a look also at the variances and standard deviations Now consider x ~ Exponential(100)

Sampling Distributions Quantitative data Expected value for x-bar is the population or process average (i.e.,  ) Expected variation in x-bar from one sample average to another is  Known as the standard error of the mean  Equal to  /√n Distribution of x-bar is approximately normal (CLT) Qualitative data: we’ll get to this another time

An Example Supposedly, WNB executive salaries equal industry on average (  = 80,000) But sample results were x-bar = $68,270 s = $18,599 If truly  = 80,000 Assume for now that  = s = What is P(x-bar < 68270)? What is P(x-bar 91730) ?

Some Answers Given assumptions about  and  Standard error:  /√n = 18599/√15 = 4800 An x-bar value of is standard errors from the supposed population average  Table probability =  Thus P(x-bar < 68270) = – = 0.7%  And P(x-bar 91730) = 1.4% Now, consider how this might be put to use in addressing the claim Bring action against WNB (false claim?) What’s the probability of doing so in error? Maybe a confidence interval estimate could be helpful...

Putting Sampling Theory to Work We need to make decisions based on characteristics of a process or population But it’s not feasible to measure the entire population or process; instead we do sampling sampling Therefore, we need to make conclusions about those characteristics based upon limited sets of observations (samples) These conclusions are inferences applying knowledge of sampling theoryinferences

Summary of Objectives Discuss the characteristics of normally distributed random variables Calculate probabilities for normal random variables Apply normal distribution concepts to practical problems Recognize other common probability distributions Summarize the concept of sampling distributions

Appendix

Random Variables Population or Process Parameters ( ,  Random Variable (x)

Sampling Population Sample Parameter Statistic

Schematic View

If Something’s Normally Distributed It’s described by  (the population/process average)population/process  (the population/process standard deviation) Histogram is symmetric Thus no skew (average = median) So P(x  ) =... ? Shape of histogram can be described by f(x) = (1/  √2  )e -[(x-  ) 2 /2  2 ] We determine probabilities based upon distance from the mean (i.e., the number of standard deviations)

A Sketch is Essential! Use to identify regions of concern Enables putting together results of calculations, lookups, etc. Doesn’t need to be perfect; just needs to indicate relative positioning Make it large enough to work with; needs annotation (probabilities, comments, etc.)