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 -4 0 -1 2 0 -1 0 2 0 1 S1 = 0 30 S2 = 105 3 i) Calculate the pooled sample covariance Spt 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.
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 -4 0 -1 2 0 -1 0 2 0 1 S1 = 0 30 S2 = 105 3 i) Calculate the pooled sample covariance Spt 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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neeDed to be solve only part a correctly in 1 hour please show neat and clean work
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- () - C)
2 0 1
-4 0
0 2 0
-1 0 3
S1 =
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 ā, which maximises the T-statistic
T =
n1 + 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 (5i – 2) .
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° = (ñm – 5s)" ((÷ +-)
Spl
n2
n1
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