2. QM 2113 - Spring 2002 Business Statistics Normally Distributed Random Variables

3. Student Objectives  Distinguish between discrete and continuous random variables  Identify from data when a variable is normally distributed  Discuss characteristics of all normally distributed random variables  Relate probability for continuous variables to “area under the curve”  Calculate probabilities associated with normally distributed random variables

4. Continuous Random Variables  Any number of values are possible between any two given values – Typically measured rather than counted – Limited practically by ability to measure  Examples – Time required to process request – Earnings per share – Distance traveled in a given amount of time – Monetary values (e.g., annual family incomes)  Now, how about examples of variables that are discrete (i.e., not continuous)?

5. Continuous Probability Distributions  Technically, like discrete distributions – List of possible values – Corresponding likelihood for each value  However, the number of possible values is infinite  Hence it’s impossible to “list” as we do with discrete distributions  Generally we use tables when working with continuous distributions

6. But Before We Go Any Further...  Any distribution is either – Empirical: we work directly with the relative frequencies – Or theoretical: we can specify probabilities as mathematical functions P(x) = x e -  / x! P(x < t ) = 1 - e - t  But we usually approximate using theoretical distributions  Most well known: normal distributions – Specific mathematical formula – Other distributions, other formulae

7. Now, For An Example  You need a car that gets at least 30 mpg  Suppose a particular model of car has been tested – Average mpg = 34 – Standard deviation = 3 mpg  Typically histograms for this type of thing look like  That is, mpg is approximately normally distributed

8. If Something’s Normally Distributed  It’s described by –  (the population/process average) –  (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)

9. Our Problem at Hand  We need a car that gets at least 30 mpg  How likely is it that this model of vehicle will meet our needs?  That is, P(x > 30) =... ? – First, sketch Number line with –Average –Also x value of concern Curve approximating histogram – Identify areas of importance – Then determine how many sigma 30 is from mu – Now use the table – Finally, put it all together

10. Comments on the Problem  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.)  Now, what do we do with the probability we’ve just determined?  Make a decision!

11. Using the Normal Table  The outside values are z-scores – That is, how many standard deviations a given x value is from the average – Use these values to look up probabilities  The body of the table indicates probabilities  Note: This is not a “z table”!  We can (and do) also work in reverse – Given a probability, determine z – Once we have z we can determine what x value corresponds to that probability

12. Some Other Exercises  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)

13. 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 <  ) = 0.5000  Tables show probabilities rounded to 4 decimal places – If z < -3.09 then probability ≈ 0.5000 – If z > 3.09 then probability ≈ 0.5000  Theoretically, P(x = a) = 0 P(30 ≤ x ≤ 35) = P(30 < x < 35)

14. Why Is This Important?  Some practical applications – Process capability analysis – Decision analysis – Optimization (e.g., ROP) – Reliability studies – Others  Most importantly, the normal distribution is the basis for understanding statistical inference  Hence, bear with this; it should be apparent soon

15. Homework  Work exercises assigned from Chapter 5  Look at Exam #3 from Spring 2000 and try to work the problems