stat 401 experimental design and analysis assist.prof.dr. r. serkan albayrak department of business...
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STAT 401 EXPERIMENTAL DESIGN AND ANALYSIS
Assist.Prof.Dr. R. Serkan AlbayrakDepartment of Business Administration
Yaşar University
Chapter 2 - Inference based on population models
Population models
Population expectations and variances
Asymptotic consistency
Testing for population differences
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Testing with the t-statistic
The normal distribution
Normally distributed data
Normally distributed means
Why? Time to remember the discussions on degrees of freedom.
z1
z2 has a bivariate normal distribution.
𝒇 (𝒛𝟏 ,𝒛𝟐 )= 𝟏
𝟐𝝅𝝈𝒛𝟏𝝈𝒛𝟐
√𝟏− 𝝆𝟐𝐞𝐱𝐩 ¿
-10-5
05
10 -10
-5
0
5
10
0.000
0.005
0.010
0.015
Bivariate Normal Distribution
z1
z2
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
x
dnor
m (x
)
If has a bivariate normal distribution then the pdf of points is a one dimensional normal distribution. This distribution is over the line because all points of the form is situated on this line.
If has a multinomial normal distribution then the pdf of points is a two dimensional normal distribution. This distribution is over the plane because all points of the form is situated on this plane. (Hard to draw)This is how one
dimension (degrees of freedom) is lost!
That means even though the points lie in a two dimensional space, the probability distribution function defined over them is basically single dimensional.
But,
The situation resembles the following: Assume we have two normally distributed random variables; and . Then the distribution of the sum their squares, i.e., does not necessarily have a Chi-squared distribution with two degrees of freedom. Why?
Consider the case where . Then which has a Chi-square distribution of one degree of freedom. Hence unless are independent has chi-square distribution with one degree of freedom.
What is distribution?
The t-distribution
That is why we divide by (n-1) in calculating sample s.d.
One sample t-test
Two-sample tests
Numerical Example (wheat again)
R-code
Comparison to the randomization test:
Randomization and t null distributions
Keep the following concepts clear:
Checking Assumptions
Checking normality
Normal scores plots
Unequal variances
Modified t-statistic
Which two-sample t-test to use?
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