91Ó°ÊÓ

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. Let \(S_{y y}=\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}\) and \(S_{x x}=\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2} .\) Show that $$\widehat{\beta}_{1}=r \sqrt{\frac{S_{y y}}{S_{x x}}}$$.b. Conditional on \(X_{i}=x_{i},\) for \(i=1,2, \ldots, n,\) show that under \(H_{0}: \rho=0\) $$\frac{\widehat{\beta}_{1} \sqrt{(n-2) S_{x x}}}{\sqrt{S_{y y}\left(1-r^{2}\right)}}$$ has a \(t\) distribution with \((n-2)\) df. c. Conditional on \(X_{i}=x_{i},\) for \(i=1,2, \ldots, n, n,\) 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.

Step by step solution

01

Understand the relationship between beta and correlations

In the context of a bivariate normal distribution, \(\widehat{\beta}_1\) is the slope coefficient that implies the change in \(Y\) for a unit change in \(X\). The correlation coefficient \(r\) represents the strength and direction of a linear relationship between the variables. The formula \(\widehat{\beta}_1 = r \sqrt{\frac{S_{yy}}{S_{xx}}}\) shows how the slope coefficient \(\widehat{\beta}_1\) can be derived from the correlation coefficient \(r\) and the ratio of the total variability of \(Y\) to that of \(X\).

02

Conditions under the null hypothesis

Under the null hypothesis \(H_0: \rho = 0\), \(Y_i\)'s are uncorrelated with \(X_i\)'s. Therefore, the ordinary least squares estimate \(\widehat{\beta}_1\) under \(H_0\) is expected to be centered around zero but has variability dependent on the data variability.

03

Derive the t-distribution under null hypothesis

Under \(H_0: \rho = 0\), the following expression converges to a t-distribution with \(n-2\) degrees of freedom:\[ \frac{\widehat{\beta}_1 \sqrt{(n-2) S_{xx}}}{\sqrt{S_{yy}(1-r^2)}} \]This shows that when there is no correlation (\(\rho = 0\)), the standardized form of \(\widehat{\beta}_1\) follows a t-distribution, reflecting its variability and allowing hypothesis testing using this distribution.

04

Special case of the t distribution

From the expression in Step 3, since \(\widehat{\beta}_1 = r \sqrt{\frac{S_{yy}}{S_{xx}}}\), inserting it into:\[ T = \frac{ r \sqrt{n-2}}{\sqrt{1-r^2}} \]also follows a t-distribution with \(n-2\) degrees of freedom as shown by the convergence derived under \(H_0\). Thus, this explicit form clearly demonstrates the same t-distribution properties as \(\widehat{\beta}_1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bivariate Normal Distribution

In statistics, a bivariate normal distribution is a joint distribution of two continuous variables, typically denoted as \((X, Y)\). This distribution is characterized by each variable having a normal distribution and the variables having a linear correlation.

When two variables possess a bivariate normal distribution, it assumes:

• Both variables have normal marginal distributions with their means \( \mu_X \) and \( \mu_Y \).
• They have variances \( \sigma_X^2 \) and \( \sigma_Y^2 \).
• There is a correlation coefficient \( \rho \), ranging from -1 to 1, indicating the strength and direction of their linear relationship.

In essence, understanding the bivariate normal distribution is crucial as it forms the backbone of many statistical analyses involving two variables. It helps to predict how a change in one variable might affect changes in another.

Correlation Coefficient

The correlation coefficient, typically represented by \( r \), quantifies the strength and direction of the linear relationship between two variables. It is a core component of statistical hypothesis testing, particularly in scenarios dealing with bivariate distributions.

Key points about the correlation coefficient:

• It ranges from -1 to 1 where -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and 0 signifies no linear relationship.
• It forms the basis for testing the null hypothesis \( H_0: \rho = 0 \), which proposes no linear relationship between \( X \) and \( Y \).
• When the correlation is zero, it implies no linear association between the variables. However, the variables could still have a non-linear relationship.

Understanding how correlation works aids in interpreting statistical results and making data-driven decisions.

T-distribution

The t-distribution is a probability distribution that is used extensively for making inferences about means and other parameters in statistics. It is particularly useful when dealing with small sample sizes or unknown population variances.

Key features of the t-distribution include:

• It resembles the normal distribution but has heavier tails, which means it is more prone to producing values far from its mean.
• The shape of the t-distribution depends on the degrees of freedom \((n-2)\). As the degrees of freedom increase, the t-distribution approaches a normal distribution.
• The t-distribution is used in hypothesis testing to determine if a given sample mean differs significantly from a population mean.

In hypothesis testing, specifically when testing for the correlation coefficient \( r \), the t-distribution helps determine if \( r \) is significantly different from zero.

Slope Coefficient

The slope coefficient \( \widehat{\beta}_1 \) estimates the change in the dependent variable \( Y \) for a one-unit change in the independent variable \( X \). It links directly to the concept of linear regression and the correlation coefficient.

Here’s why the slope coefficient is important:

• \( \widehat{\beta}_1 \) shows how sensitive the dependent variable is to changes in the independent variable. A larger magnitude implies a stronger relationship.
• In statistical hypothesis testing, under the null hypothesis \( H_0: \rho = 0 \), the estimate for the slope coefficient should lie around zero, suggesting no relationship.
• The expression \( \widehat{\beta}_1 = r \sqrt{\frac{S_{yy}}{S_{xx}}} \) ties the slope coefficient to the correlation coefficient \( r \) and variability of \( X \) and \( Y \).

The slope coefficient is crucial in regression analysis as it conveys the direction and magnitude of linear relationships between variables.