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?