2. QM 2113 - Spring 2002 Business Statistics Exercises with Normally Probabilities

3. Student Objectives  Calculate probabilities associated with normally distributed random variables  Apply normal distribution calculations to various decision making situations

4. First, an Announcement  Visits this week by major firms – Cardinal Health – Acxiom  Cardinal Health (Kathy White) – Who are they? – Why do we care? – When/where  Acxiom – Who are they and what do they do? – When/where

5. Working with Normal Distributions  First, sketch – Number line with Average (i.e.,  ) Also x value of concern – Curve approximating histogram  Identify areas of importance  Then determine how many standard deviations x value is from   Now use the table  Finally, put it all together

6. Mechanics: Some Calculation 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)

7. Recall About 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

8. 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)

9. 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

10. Homework  Rework (as necessary) exercises assigned from Chapter 5  Work problems on Exam #3 from Spring 2000  Review for midterm exam