Question 1. a) Consider an experiment which measures the value of three different vari- ables on a unit. This experiment was conducted twice, giving two samples (both with 22 observations each), with sample means 0. 1 -1 あ= -2 and sample covariance matrices 201 -4 0 -1 S1 = 0 2 0 -1 0 3 030 S2 = 105 i) Calculate the pooled sample covariance Spi for this data; ii) Calculate the corresponding Hotelling's T2-statistic and thus, conclude if the null hypothesis Ho should be rejected at the 1% significance level by comparison to a critical value from an F-table. b) If the null hypothesis in the two-sample T2-test is rejected, i.e. the two population means are not equal, we can determine which variable contributed the most to this rejection by finding the linear transformation coefficient vector a, which maximises the T-statistic T = ni +n2 aTSā nin2 where Spl is the pooled sample covariance. It can be shown that the coefficient vector which maximises this statistic is the so-called 'discriminant function' ā = S, (5 – 52) . Using the discriminant function, show that the square of the maximised T-statistic is nothing other than the original Hotelling's T2 statistic for two samples, i.e. -1 T° = (m – i»)" ((, +) s.)"(5n – 5s)- 1 Sp (51 – 52) .
Question 1. a) Consider an experiment which measures the value of three different vari- ables on a unit. This experiment was conducted twice, giving two samples (both with 22 observations each), with sample means 0. 1 -1 あ= -2 and sample covariance matrices 201 -4 0 -1 S1 = 0 2 0 -1 0 3 030 S2 = 105 i) Calculate the pooled sample covariance Spi for this data; ii) Calculate the corresponding Hotelling's T2-statistic and thus, conclude if the null hypothesis Ho should be rejected at the 1% significance level by comparison to a critical value from an F-table. b) If the null hypothesis in the two-sample T2-test is rejected, i.e. the two population means are not equal, we can determine which variable contributed the most to this rejection by finding the linear transformation coefficient vector a, which maximises the T-statistic T = ni +n2 aTSā nin2 where Spl is the pooled sample covariance. It can be shown that the coefficient vector which maximises this statistic is the so-called 'discriminant function' ā = S, (5 – 52) . Using the discriminant function, show that the square of the maximised T-statistic is nothing other than the original Hotelling's T2 statistic for two samples, i.e. -1 T° = (m – i»)" ((, +) s.)"(5n – 5s)- 1 Sp (51 – 52) .
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:Question 1. a) Consider an experiment which measures the value of three different vari-
ables on a unit. This experiment was conducted twice, giving two samples (both with
22 observations each), with sample means
-1
-2
-2
and sample covariance matrices
s- () --()
2 0 1
S1 =
030
0 2
-1 0 3
S2 =
105
i) Calculate the pooled sample covariance Spi for this data;
ii) Calculate the corresponding Hotelling's T2-statistic and thus, conclude if the null
hypothesis Ho should be rejected at the 1% significance level by comparison to a
critical value from an F-table.
b) If the null hypothesis in the two-sample T2 -test is rejected, i.e. the two population means
are not equal, we can determine which variable contributed the most to this rejection by
finding the linear transformation coefficient vector ā, which maximises the T-statistic
T =
ni + n2
nin2
where Spi is the pooled sample covariance. It can be shown that the coefficient vector
which maximises this statistic is the so-called 'discriminant function'
S (51 – 72) .
ā =
Using the discriminant function, show that the square of the maximised T-statistic is
nothing other than the original Hotelling's T2 statistic for two samples, i.e.
-1
T° = (n – 75)" ((- +-)
Spl
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