(In some of the exercises that follow, we must

QUESTION:

Let \(X\) and \(Y\) denote the weights in grams of male and female common gallinules, respectively. Assume that \(X\) is \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(Y\) is \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\).

(a) Given \(n=16\) observations of \(X\) and \(m=13\) observations of \(Y\), define a test statistic and a critical region for testing the null hypothesis \(H_{0}: \mu_{X}=\mu_{Y}\) against the one-sided alternative hypothesis \(H_{1}: \mu_{X}>\mu_{Y}\). Let $\alpha=0.01$. (Assume that the variances are equal.)

(b) Given that \(\bar{x}=415.16, s_{x}^{2}=1356.75, \bar{y}=347.40\), and \(s_{y}^{2}=692.21\), calculate the value of the test statistic and state your conclusion.

(c) Although we assumed that \(\sigma_{X}^{2}=\sigma_{Y}^{2}\), let us say we suspect that that equality is not valid. Thus, use the test proposed by Welch.

Equation Transcription:

Text Transcription:

X  

Y  

N(mu_X, sigma_X^2)

N(mu_Y, sigma_Y^2)  

n=16  

m=13  

H_0:mu_X=mu_Y  

H_1:mu_X>mu_Y

alpha=0.01  

Bar x=415.16, s_x^2=1356.75, bar y=347.40

s_y^2=692.21

sigma_X^2=sigma_Y^2