Let ( X, Y ) have a bivariate normal distribution. A test

Chapter 11, Problem 105SE

(choose chapter or problem)

Let (X, Y ) have a bivariate normal distribution. A test of \(H_{0}: \rho=0\) against \(H_{a}: \rho \neq 0\) can be derived as follows.

a.  \(\text { Let } S_{y y}=\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2} \text { and } S_{x x}=\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2} \text {. Show that }\)

\(\hat{\beta}_{1}=r \sqrt{\frac{S_{y y}}{S_{x x}}}\)

b.  \(\text { Conditional on } X_{i}=x_{i} \text {, for } i=1,2, \ldots, n, \text { show that under } H_{0}: \rho=0\)

\(\frac{\hat{\beta}_{1} \sqrt{(n-2) S_{x x}}}{\sqrt{S_{y y}\left(1-r^{2}\right)}}\)

\(\text { has a } t \text { distribution with }(n-2) \text { df. }\)

c.  \(\text { Conditional on } X_{i}=x_{i} \text {, for } i=1,2, \ldots, n \text {, conclude that }\)

\(T=\frac{r \sqrt{n-2}}{\sqrt{1-r^{2}}}\)

has a t distribution with (n−2) df, under \(\(H_{0}: \rho=0\)\). Hence, conclude that T has the same distribution unconditionally.

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Text Transcription:

H0: \rho=0      

Ha:\rho \neq 0

Let Syy=\sum _i=1n(yi-y)2 and Sxx= \sum_i=1n(xi-x)2 show that      

\hat \beta=r \sqrt{\frac SyySxx

Conditional on Xi=xi, for i=1,2,...,n, show that under H0:\rho=0          

\frac \hat \beta_1 \sqrt(n-2) S_x x\sqrt{S_{y y(1-r^2)

has a t distribution with(n-2)df  

Conditional on Xi=xi, for i=1,2,...,n, conclude that      

T=\frac r \sqrt n-2 \sqrt 1-r^2

H0:  \rho=0