Question

Für einen zufälligen Vektor \((X, Y)\) wurden 11 Werte \(X=x_{i}\) eingestellt und dazu die Werte \(y_{i}\) der bedingten Zufallsgrößen \(\left(Y \mid X=x_{i}\right)\) gemessen: \begin{tabular}{cccccccccccc} \hline\(i\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline\(x_{i}\) & \(-2\) & \(-1\) & \(-1\) & 0 & 1 & 2 & 2 & 3 & 4 & 4 & 5 \\ \hline\(y_{i}\) & \(-0,3\) & 0,8 & 1,2 & 1,9 & 3,0 & 4,5 & 3,9 & 5,4 & 6,1 & 6,2 & 6,8 \\ \hline \end{tabular} \((Y \mid X=x)\) sei vom Typ \(N\left(\beta_{0}+\beta_{1} x, \sigma\right)\) (a) Man bestimme mittels der angegebenen Stichprobe Schätzwerte für die \(\mathrm{Re}\) gressionskoeffizienten \(\beta_{0}, \beta_{1}\) sowie für \(\sigma^{2}\). (b) Man gebe Konfidenzintervalle für \(\beta_{0}, \beta_{1}\) zum Konfidenzniveau \(\alpha=0,05\) an. (c) Man prüfe die Hypothesen \((\alpha=0,05)\) \(H_{0}: \beta_{0}=2\) gegen \(\quad H_{1}: \beta_{0} \neq 2\) und \(H_{0}: \beta_{1}=1\) gegen \(\quad H_{1}: \beta_{0} \neq 1\)

Short answer

\(\beta_0 \approx -0.357, \beta_1 \approx 2.761\). Reject \(H_0: \beta_0 = 2\) and \(H_0: \beta_1 = 1\).

Step-by-step solution

Calculate the Means

Calculate the mean of the values of \( x_i \) and \( y_i \). The mean of \( x_i \) is given by \[ \bar{x} = \frac{-2 + (-1) + (-1) + 0 + 1 + 2 + 2 + 3 + 4 + 4 + 5}{11} = 1.3636 \]The mean of \( y_i \) is given by \[ \bar{y} = \frac{-0.3 + 0.8 + 1.2 + 1.9 + 3.0 + 4.5 + 3.9 + 5.4 + 6.1 + 6.2 + 6.8}{11} = 3.4273 \]

Calculate the Regression Coefficients

Next, calculate the regression coefficients \( \beta_1 \) and \( \beta_0 \) using the formulas:\[ \beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]\[ \beta_0 = \bar{y} - \beta_1 \bar{x} \].After substituting the values and calculating, we get\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 64.2545 \]\[ \sum (x_i - \bar{x})^2 = 23.2727 \]Hence \[ \beta_1 = \frac{64.2545}{23.2727} \approx 2.7610 \]\[ \beta_0 = 3.4273 - (2.7610)(1.3636) \approx -0.3571 \].

Estimate the Variance \(\sigma^2\)

The variance \( \sigma^2 \) can be estimated using:\[\sigma^2 = \frac{1}{n-2}\sum (y_i - \hat{y}_i)^2\]where \( \hat{y}_i = \beta_0 + \beta_1 x_i \).Calculate \( y_i - \hat{y}_i \) for each \( x_i \) and sum the squares of these differences. After calculations,\[ \sigma^2 \approx \frac{1}{9} \times 3.0509 = 0.33899 \].

Confidence Intervals for \(\beta_0\) and \(\beta_1\)

The confidence interval for \( \beta \) is given by\[\beta \pm t^* \times \text{SE}(\beta)\]where \( t^* \) is the critical value from the t-distribution at \( \alpha = 0.05 \). To find the standard errors, use:\[ \text{SE}(\beta_0) = \sqrt{\sigma^2 \left(\frac{1}{n} + \frac{\bar{x}^2}{\sum (x_i-\bar{x})^2}\right)} \]\[ \text{SE}(\beta_1) = \sqrt{\frac{\sigma^2}{\sum (x_i-\bar{x})^2}} \].After calculation\( t^* = 2.262 \) (for 9 degrees of freedom):\[ \beta_0: -0.3571 \pm 2.262 \times 0.3678 \approx (-0.959, 0.245) \]\[ \beta_1: 2.7610 \pm 2.262 \times 0.1205 \approx (2.479, 3.043) \].

Hypothesis Testing

For hypothesis testing:1. Calculate the test statistics: \[ t_{\beta_0} = \frac{\beta_0 - 2}{\text{SE}(\beta_0)} \approx \frac{-0.3571 - 2}{0.3678} \approx -6.4139 \] \[ t_{\beta_1} = \frac{\beta_1 - 1}{\text{SE}(\beta_1)} \approx \frac{2.7610 - 1}{0.1205} \approx 14.6224 \].2. Compare these values against the critical value (2-tailed) for 9 degrees of freedom at \( \alpha = 0.05 \), which is \( t^* = 2.262 \).Both \( t_{\beta_0} \) and \( t_{\beta_1} \) lead us to reject the null hypotheses \( H_0: \beta_0=2 \) and \( H_0: \beta_1=1 \) respectively.

Key concepts

Zufallsvektor

A random vector, or "Zufallsvektor" in German, is simply a set of random variables that can be represented in vector form. Typically, these random variables can be correlated, meaning the outcome of one may potentially influence the outcome of another. In the context of regression analysis, two variables, such as \(X\) and \(Y\), are considered. These variables may represent different measurable quantities, with their joint distribution defining the characteristics of the random vector.

When performing regression analysis with a random vector, you aim to understand how one variable influences another. For example, you may be interested in how variable \(X\) influences variable \(Y\). This relationship is often modeled linearly, where \(Y\), given \(X\), follows a normal distribution \(N(\beta_{0}+\beta_{1} x, \sigma)\). This setup is the foundation of the exercise, where we calculate regression coefficients to quantify the nature of the relationship between \(X\) and \(Y\).

• Random vectors can involve both discrete and continuous distributions.
• They are defined by their joint distribution function.
• Used to model real-world phenomena where variables are interdependent.

Konfidenzintervall

Confidence intervals, or "Konfidenzintervalle," are essential in statistical analysis, providing a range of values that are likely to contain the true parameter of interest. These intervals give us an estimate of how precise our calculated regression coefficients, like \(\beta_0\) and \(\beta_1\), actually are.

In regression analysis, after estimating these coefficients based on sample data, a confidence interval can be constructed around each estimate to express the uncertainty due to sampling variability. A 95% confidence interval indicates that if we were to take many samples and build confidence intervals in the same way, 95% of them would expectedly contain the true parameter value.

• Important for understanding the reliability of an estimate.
• Interpreted as a range where the true parameter is "likely" to be found.
• Depends on the size, standard deviation, and level of confidence \(\alpha\).

Hypothesentest

Hypothesis testing, known as "Hypothesentest" in German, is a fundamental aspect of statistical analysis. It involves making an initial assumption (the null hypothesis \(H_0\)) and testing whether enough statistical evidence exists to reject this in favor of an alternative hypothesis \(H_1\).

The primary goal of hypothesis tests in regression is to determine whether the estimated coefficients (like \(\beta_0\) and \(\beta_1\)) significantly differ from a hypothesized value. For the given exercise, two hypotheses - involving \(\beta_0 = 2\) and \(\beta_1 = 1\) - are tested against alternative claims. By calculating a test statistic and comparing it to a critical value from a t-distribution, we determine the plausibility of these hypotheses.

• Hypothesis tests help in deciding the validity of assumptions made.
• The test result leads to rejecting or failing to reject the null hypothesis.
• Significance level \(\alpha\) denotes the probability of wrongly rejecting a true null hypothesis.

Regressionskoeffizienten

Regression coefficients, or "Regressionskoeffizienten," are the parameters \(\beta_0\) and \(\beta_1\) in the linear regression equation \(Y = \beta_{0} + \beta_{1}X + \epsilon\), where \(\epsilon\) is the error term. These coefficients are vital as they quantify the relationship between the independent variable \(X\) and the dependent variable \(Y\).

The coefficient \(\beta_1\) represents the slope, indicating how much \(Y\) changes for a one-unit change in \(X\). Conversely, \(\beta_0\) is the intercept, representing the expected value of \(Y\) when \(X = 0\). Calculating these coefficients involves minimizing the sum of squared differences between the observed \(y_i\) values and the predicted \(\hat{y}_i\) values.

• \(\beta_1\): Describes the direction and steepness of the relationship.
• \(\beta_0\): Provides a baseline value for \(Y\) with no \(X\) influence.
• Derived using the method of least squares to best fit the data points.