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
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
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