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Business Statistics

This chapter delves into the significance of the bell-shaped curve, also known as the normal or Gaussian curve, as it relates to data sets in statistics. It explains the Central Limit Theorem, which states that the distribution of sample means approximates normality with an infinite number of samples. The chapter covers key concepts like density functions, the relationship between the area under the curve and probabilities, and various statistical problems involving normal distributions, including practical applications such as insurance sales and airplane door heights.

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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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