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Quantitative Methods for Lawyers - Class #16 - Students "t" Distribution - Professor Daniel Martin Katz
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Quantitative Methods for Lawyers - Class #16 - Students "t" Distribution - Professor Daniel Martin Katz

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  • 1. Quantitative Methods for Lawyers Students “t” Distribution Class #16 professor daniel martin katz computationallegalstudies.com @ computational
  • 2. Students “T” Distribution
  • 3. Students “T” Distribution v. Normal Distribution Let X1, X2,..., Xn be drawn from N ( μ,σ ) We have learned that is then distributed Standard Normal If we know σ than we can use Z Scores But typically - we do not actually know σ
  • 4. Students “T” Distribution Student “T” Distribution is preferred statistic for dealing with continuous data Sample sizes are sometimes small, and often we do not know the standard deviation of the population. When either of these problems occur, statisticians rely on “t” distribution
  • 5. Students “T” Distribution The t distributions were discovered by William S. Gosset in 1908. Goal for Gosset: Determine the Likelihood that any particular sample represented the true quality of the entire product Comparing the Mean of Population and Mean of a Given Sample
  • 6. Students “T” Distribution Gosset was a statistician employed by the Guinness brewing company which had stipulated that he not publish under his own name. He therefore wrote under the pen name “Student.”
  • 7. Students “T” Distribution The t distribution should NOT be used with small samples from populations that are NOT approximately normal The particular form of the t distribution is determined by its degrees of freedom
  • 8. Students “T” Distribution NOTE: T-Distribution Converges to the Normal Distribution A Student's t distribution converges to a normal distribution when the number of degrees of freedom N becomes large (converges to infinity). http://www.nku.edu/~longa/stats/taryk/TDist.html
  • 9. Students “T” Distribution A Student's t distribution when the N is small Otherwise, use Normal and “Z Scores” What is “Small” in this context? If the sample is small, n < 30, we use t and if the sample is large, n ≥ 30, we use z.
  • 10. Students “T” Distribution http://www.nku.edu/~longa/stats/ taryk/TDist.html
  • 11. Students “T” Distribution Different Forms Single Sample T Test Problem Comparing the Means of Two Samples
  • 12. Students “T” Distribution Acme Corporation manufactures light bulbs. The CEO claims that an average Acme light bulb lasts 300 days. A researcher randomly selects 15 bulbs for testing. The sampled bulbs last an average of 290 days, with a standard deviation of 50 days. If the CEO’s claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than 290 days?
  • 13. Students “T” Distribution Acme Corporation manufactures light bulbs. The CEO claims that an average Acme light bulb lasts 300 days. A researcher randomly selects 15 bulbs for testing. The sampled bulbs last an average of 290 days, with a standard deviation of 50 days. If the CEO’s claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than 290 days? This is Single Sample T Test Problem
  • 14. Students “T” Distribution
  • 15. Students “T” Distribution P Value
  • 16. Students “T” Distribution http://stattrek.com/Tables/T.aspx
  • 17. Example From Our Book Involving Damage Awards Awards in Rest of State Awards in Bloom County 235,000 175,000 750,000 230,000 450,000 150,000 1,000,060 910,000 150,000 220,000 130,000 170,000 234,000 450,000 890,000 101,000 120,000 560,000 321,000 456,000 102,000 30,000 793,000 250,900 862,000 673,000 463,000 54,000 39,000 687,000 260,800 682,000 3,514,000 67,000 356,000 13,000 42,000 4,000 402,000 943,000 961,600 630,000 398,800 52,000 976,500 540,000 N = 21 N = 25
  • 18. Are Damage Awards in Bloom Awards in Rest of State Awards in Bloom County 235,000 175,000 750,000 230,000 450,000 150,000 1,000,060 910,000 150,000 220,000 130,000 170,000 234,000 450,000 890,000 101,000 120,000 560,000 321,000 456,000 102,000 30,000 793,000 250,900 862,000 673,000 463,000 54,000 39,000 687,000 260,800 682,000 3,514,000 67,000 356,000 13,000 42,000 4,000 402,000 943,000 961,600 630,000 398,800 52,000 976,500 540,000 N = 21 N = 25 County Excessive? H0: There is No Difference Between the Mean Damage Award in Bloom County and the Mean Damage Award in the Rest of the State This is a Two Sample Problem
  • 19. H0: There is No Difference Between the Mean Damage Award in Bloom County and the Mean Damage Award in the Rest of the State Num of Obs. Mean Std. Dev. GROUP 1 Rest of State 21 $371,621 $289,823 GROUP 2 Bloom County 25 $547,784 $703,314
  • 20. There are Various Approaches You Might Take On the Left I Manually Entered the Data is in Excel Here is the Data Set With 2 Variables: Award = Award Amount in Dollars Bloom = Indicator Variable ( where 1 = award in Bloom County ) ( where 0 = award in rest of the State) You can then load this into Then you can calculate the two mean test
  • 21. Use an online t-test calculator http://www.graphpad.com/quickcalcs/ttest1.cfm