(In some of the exercises that follow, we must

Chapter 8, Problem 1E

(choose chapter or problem)

The botanist in Example 8.2-1 is really interested in testing for synergistic interaction. That is, given the two hormones gibberellin \(\left(\mathrm{GA}_{3}\right)\) and indoleacetic acid ((\mathrm{IAA})\), let \(X_{1}\) and \(X_{2}\) equal the growth responses (in \(\mathrm{mm}\) ) of dwarf pea stem segments to \(\mathrm{GA}_{3}\) and \(\mathrm{IAA}\), respectively and separately. Also, let \(X=X_{1}+X_{2}\) and let \(Y\) equal the growth response when both hormones are present. Assuming that \(X\) is \(N\left(\mu_{X}, \sigma^{2}\right)\) and \(Y\) is \(N\left(\mu_{Y}, \sigma^{2}\right)\), the botanist is interested in testing the hypothesis \(H_{0}: \mu_{X}=\mu_{Y}\) against the alternative hypothesis of synergistic interaction \(H_{1}: \mu_{X}<\mu_{Y}\).

(a) Using \(n=m=10\) observations of \(X\) and \(Y\), define the test statistic and the critical region. Sketch a figure of the \(t\) pdf and show the critical region on your figure. Let \(\alpha=0.05\).

(b) Given \(n=10\) observations of \(X\), namely,

                            2.1 2.6 2.6 3.4 2.1 1.7 2.6 2.6 2.2 1.2

                   and \(m = 10\) observations of \(Y\), namely,

 

                          3.5 3.9 3.0 2.3 2.1 3.1 3.6 1.8 2.9 3.3

calculate the value of the test statistic and state your conclusion. Locate the test statistic on your figure.

(c) Construct two box plots on the same figure. Does this confirm your conclusion?

Equation Transcription:

 

 

 

 

 

 

 

 

 

 

Text Transcription:

(GA_3)  

X_1  

X_2  

mm  

GA_3

IAA

X=X_1+X_2  

Y

X  

N(mu_X, sigma^2)  

N(mu_Y, sigma^2)

H_0:mu_X=mu_Y  

H_1:mu_X<mu_Y

n=m=10  

t  =0.05

n=10

m=10