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Business Statistics

Business Statistics. Chapter 8. Bell Shaped Curve. Describes some data sets Sometimes called a normal or Gaussian curve – I’ll use normal. Central Limit Theorem.

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Business Statistics

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  1. Business Statistics Chapter 8

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

  3. 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

  4. 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 (s and m)

  5. Approximately 68% of means within +/- 1σ 95% of means within +/- 2σ 99.7% within +/- 3σ .3413 .3413 .1359 .1359 .0215 .0215 m-s m m+s m-2s m+2s Area under curve = 1.0 (100% of the probability)

  6. Some Problems • A normal distribution has parameters s = 20 and m = 100 • What fraction of values will fall: • Between 60 and 100? • Between 120 and 140? • Below 50?

  7. What fraction of values will fall: • Between 60 and 100? .1359 + .3413 = .4772 = (.4772/1.0000) = 47.72% .3413 .1359 μ-3σ μ-2σ μ-1σ μ +1σ μ +2σ μ +3σ

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

  9. What fraction of values will fall: • Below 50? .5000 or 50% .0215 .3413 + .1359 = .4772 μ-3σ μ-2σ μ-1σ μ +1σ μ +2σ μ +3σ

  10. Some Problems (this time with EXCEL) • A normal distribution has parameters s = 20 and m = 100 • What fraction of values will fall: • Between 60 and 100? • Between 120 and 140? • Below 50? • Above 75?

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

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

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

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

  15. Some Problems • A normal distribution has parameters s = 20 and m = 100 • 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

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

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

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

  19. 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

  20. 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?

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