Quantitative Methods for Lawyers - Class #13 - Students "t" Distribution - Professor Daniel Martin Katz

1. Quantitative
Methods
for
Lawyers Class #13
Students “t” Distribution
@ computational
computationallegalstudies.com
professor daniel martin katz danielmartinkatz.com
lexpredict.com slideshare.net/DanielKatz

2. Students “T”
Distribution

3. Students “T” Distribution
v. Normal Distribution
is then distributed Standard Normal
Let X1, X2,..., Xn be drawn from N ( μ,σ )
We have learned that
But typically - we do not actually know σ
If we know σ than we can use Z Scores

4. Student “T” Distribution is preferred statistic for dealing with
continuous data
Students “T”
Distribution
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. The t distributions were discovered by William S. Gosset
in 1908.
Students “T”
Distribution
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. 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.”
Students “T”
Distribution

7. The t distribution should NOT be used with small
samples from populations that are NOT approximately
normal
Students “T”
Distribution
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”
If the sample is small, n < 30, we use t and if
the sample is large, n ≥ 30, we use z.
What is “Small” in this context?

10. Students “T”
Distribution
http://www.nku.edu/~longa/stats/
taryk/TDist.html

11. Students “T”
Distribution
Different Forms
Comparing the Means
of Two Samples
Single Sample T Test Problem

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
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
Awards in Rest of State Awards in Bloom County
N = 21
N = 25

18. 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
Awards in Rest of State Awards in Bloom County
N = 21
N = 25
Are Damage Awards in Bloom
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. 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)
There are Various Approaches
You Might Take
You can then load this into
On the Left I Manually Entered
the Data is in Excel
Then you can calculate the two mean test

22. Use an online t-test calculator
http://www.graphpad.com/quickcalcs/ttest1.cfm

23. Daniel Martin Katz
@ computational
computationallegalstudies.com
lexpredict.com
danielmartinkatz.com
illinois tech - chicago kent college of law@