1. Business Statistics Chapter 8

2. Bell Shaped Curve Describes some data sets Sometimes called a normal or Gaussian curve – I’ll use normal

4. Central Limit Theorem The central limit theorem states that when an infinite number of successive random samples are taken from a population, the distribution of sample means calculated for each sample will become approximately normally distributed

5. A Family of Curves Density Functions The area under the curve represents the population Probabilities can be determined by viewing the area under the curve A normal density is specified by its standard deviation and mean (  and 

8.      Approximately 68% of means within +/- 1σ 95% of means within +/- 2σ 99.7% within +/- 3σ Area under curve = 1.0 (100% of the probability).3413.1359.0215

9. Some Problems A normal distribution has parameters  = 20 and  What fraction of values will fall: Between 60 and 100? Between 120 and 140? Below 50?

10. What fraction of values will fall: Between 60 and 100? μ-1σ μ-2σ μ-3σ μ +1σ μ +2σμ +3σ.3413.1359.1359 +.3413 =.4772 = (.4772/1.0000) = 47.72%

11. What fraction of values will fall: Between 120 and 140? μ-3σ μ-2σ μ-1σ μ +1σ μ +2σμ +3σ.1359

12. What fraction of values will fall: Below 50? μ-3σ μ-2σ μ-1σ μ +1σ μ +2σμ +3σ.5000 or 50%.3413 +.1359 =.4772.0215

13. Some Problems (this time with EXCEL) A normal distribution has parameters  = 20 and  What fraction of values will fall: Between 60 and 100? Between 120 and 140? Below 50? Above 75?

14. NORMDIST(100,100,20,TRUE)-NORMDIST(60,100,20,TRUE) =.5 -.02275 =.47725 What fraction of values will fall: Between 60 and 100?

15. NORMDIST(140,100,20,TRUE)-NORMDIST(120,100,20,TRUE) What fraction of values will fall: Between 120 and 140?

16. NORMDIST(50,100,20,TRUE) What fraction of values will fall: Below 50?

17. 1 - NORMDIST(75,100,20,TRUE) What fraction of values will fall: Above 75?

18. Some Problems A normal distribution has parameters  = 20 and  What value will: 50% of the values fall below? 20% of the values fall above? 10% of the values fall above? What range contains 95% of the values

19. 50% NORMINV(.5,100,20)

20. 20% 80% NORMINV(.2,100,20)

21. 2.5% 95% NORMINV(.975,100,20) and NORMINV(.025,100,20) 2.5%

22. Insurance Sales The Great Buffalo Insurance company has 3,000 agents nationwide Annual sales per agent average $1,500,000 with a standard deviation of $350,000 The sales manager wishes to set a goals such that 25%, 10%, and 2% of the agents will exceed the goals The distribution of sales in normal

23. Height of Airplane Doors Airplane passenger doors are 6 feet in height. Passenger heights have a normal distribution with m = 5’6” and s = 6” What percentage of passengers will need to duck? How high should the doors be made so that only 10% of the passengers must duck?