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world:quiz1
Day 1
We observe \((x_i,y_i)\) for \(i \in [1:n]\) where \(y_i=\beta_1 + \beta_2 x_i+\epsilon_i\) for \(i \in [1:n]\). We assume that there exists \(i\neq j\), such that \(x_i\neq x_j\) and \(\mathbb{E}[\epsilon_i]=0\) and \(\mathbb{C}\mathrm{ov}(\epsilon_i,\epsilon_j)=\sigma^2 \mathbf{1}(i \neq j)\). We use the notation: \(\bar x=\sum_{i=1}^n x_i/n\) and \(\bar y=\sum_{i=1}^n y_i/n\). Which of the following statements are always true:
| Number | statement | answer |
|---|---|---|
| 1 | \(\sum_{i=1}^n (x_i-\bar x)=0\) | TRUE |
| 2 | \((\sum_{i=1}^n x_i)-\bar x=0\) | FALSE |
| 3 | The regression coefficient estimator \(\hat \beta_2\) is given by: \(\hat \beta_2=\frac{\sum_{i=1}^n x_i(y_i-\bar y)}{\sum_{i=1}^n (x_i-\bar x)^2}\) | TRUE |
| 4 | The regression coefficient estimator \(\hat \beta_2\) is given by: \(\hat \beta_2=\frac{(\sum_{i=1}^n x_i y_i/n) -\bar x \bar y}{\sum_{i=1}^n x_i^2/n-(\bar x)^2}\) | TRUE |
| 5 | The regression coefficient estimator \(\hat \beta_2\) is given by: \(\hat \beta_2=\frac{(\sum_{i=1}^n x_i y_i) -\bar x \bar y}{(\sum_{i=1}^n x_i^2)-(\bar x)^2}\) | FALSE |
| 6 | The regression line is defined by the equation \(y=\beta_1 + \beta_2 x\). | FALSE |
| 7 | The regression line is defined by the equation \(y=\hat \beta_1 +\hat \beta_2 x\). | TRUE |
| 8 | Assume that the regression line is \(y=5 x+4\). Assume that \(n=10\), \(\sum_{i=1}^n x_i=30\). Then, we obtain that \(\sum_{i=1}^n y_i=19\). | FALSE |
| 9 | Assume that the regression line is \(y=5 x+4\). Assume that \(n=10\), \(\sum_{i=1}^n x_i=30\). Then, we obtain that \(\sum_{i=1}^n y_i=190\). | TRUE |
| 10 | Assume that the regression line is \(y=5 x+4\). Assume that \(n=10\), \(\sum_{i=1}^n x_i=30\). Then, we obtain that \(\sum_{i=1}^n y_i=154\). | FALSE |
world/quiz1.txt · Last modified: 2025/09/30 19:02 by rdouc
