Linear Statistical Models, Quiz 1

January 2025 term, 23 Feb 2025, Set QIM2

Learning mode

Question 1

+30−0Single correct

A linear regression model $\hat{y} = a_0 + a_1 x$ is fit for the cars dataset. The dataset has 50 observations with 2 variables dist and speed. Study the code and answer the following subquestions:

1data(cars)
2y <- cars$dist
3x <- cars$speed
4
5car_mod <- lm(y ~ x, data = cars)
6coef(car_mod)
7
8## Output
9> coef(car_mod)
10(Intercept)           x
11 -17.579095    3.932409

(a) Find the best fit line $\hat{y}$ . [2 Marks]

(b) Find the residual corresponding to the data point $(x, y) = (4, 2)$ . [3 Marks]

Let $X_1, X_2, \ldots, X_n$ be i.i.d. samples from a normal distribution with mean $\mu$ and variance $\sigma^2$ . Consider the sample mean and sample variance as

$\overline{X} = \frac{1}{n}\sum_{i=1}^{n} X_i, \qquad\qquad S^2 = \frac{1}{(n-1)}\sum_{i=1}^{n}(X_i - \overline{X})^2,$

then which of the following are true? Select all that apply [3 Marks]

(a) $\overline{X} \sim \text{Normal}(\mu, \sigma^2/n)$

(b) $\overline{X} \sim \text{Normal}(\mu, n\sigma^2)$

(d) $\dfrac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t_n$

(e) $\dfrac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t_{(n-1)}$

Consider the first 10 rows of the iris dataset as below:

1head(iris, 10)
2
3## output
4   Sepal.Length Sepal.Width Petal.Length Petal.Width Species
51           5.1         3.5          1.4         0.2  setosa
62           4.9         3.0          1.4         0.2  setosa
73           1.0         1.0          1.0         1.0    <NA>
84           4.6         3.1          1.5         0.2  setosa
95           5.0         3.6          1.4         0.2  setosa
106           5.4         3.9          1.7         0.4  setosa
117           2.0         2.0          2.0         2.0    <NA>
128           5.0         3.4          1.5         0.2  setosa
139           3.0         3.0          3.0         3.0    <NA>
1410          4.9         3.1          1.5         0.1  setosa

(a) What will be the output of iris[7, 5]? [1 Mark]

(b) What will be the output of the following code snippet? [2 Marks]

1iris$Sepal.Length[c(1, 4, 7)] = c(5, 5, 5)
2iris[c(1, 4, 5), ]

1var(iris$Sepal.Length)
2[1] 0.8636282

Then, find the population variance of the variable Sepal.Length. [2 Marks]

Consider the linear model :

$y_{ij} = \alpha + \beta x_{ij} + \gamma_i + \epsilon_{ij} \quad ; \quad 1 \leq j \leq n_i \, , \, 1 \leq i \leq 3,$

where :

$\gamma_i \sim N(0, \sigma^2_\gamma)$ is a random effect for group $i$ , independent of $\epsilon_{ij}$ .
$\epsilon_{ij} \sim N(0, \sigma^2_\epsilon)$ are independent errors.
$x_{ij}$ is a known covariate.

Find the distribution of $y_{ij}$ . [2 Marks]

Suppose you fit a linear regression model to the following data:

$\vec{x}$	$\vec{y}$
1	2
2	4
3	5
4	4
5	5

and the estimated regression equation is : $\hat{\vec{y}} = 2.2 + 0.6x$ . If the prediction errors be defined by $\vec{e}_i = \vec{y}_i - \hat{\vec{y}}_i$ , then calculate the value of $\sum_{i=1}^{n}(\vec{e}_i)^2$ . [4 Marks]

Let $(\vec{Y}, X\vec{\beta}, \sigma^2 I_n)$ be a linear model and $p \in \mathbb{R}^m$ . Prove that $p^T\vec{\beta}$ is estimable $\Leftrightarrow p \in C(X^T)$ . [4 Marks]
Consider a model:

$\begin{aligned} y_1 &= 2\beta_1 + 2\beta_2 + 2\beta_3 + 2\beta_4 + \epsilon_1 \,; \\ y_2 &= 2\beta_1 + 2\beta_3 - 2\beta_2 - 2\beta_4 + \epsilon_2 \,; \\ y_3 &= 2\beta_1 + 2\beta_2 - 2\beta_3 - 2\beta_4 + \epsilon_3 \,; \\ y_4 &= 2\beta_1 + 2\beta_4 - 2\beta_2 - 2\beta_3 + \epsilon_4 \,, \end{aligned}$

where $\beta_i \in R$ and $\epsilon_i$ 's are uncorrelated random variables with variance $\sigma^2$ ; $i = 1, 2, 3, 4$ .

(a) If we want to rewrite the model as $(\vec{Y}, X\vec{\beta}, I_{4\times4})$ , then find $\vec{Y}, X, \vec{\beta}$ and $\vec{\epsilon}$ . [2Marks]

(b) Define Normal equations and using normal equations find the least square estimates of $\beta_1, \beta_2, \beta_3$ and $\beta_4$ . [5 Marks]

A.
I have written answers on the answer sheets
B.
Not applicable

1. A linear regression model $\hat{y} = a_0 + a_1 x$ is fit for the `cars` dataset. The dataset has 50 observations with 2 variables `dist` and `speed`. Study the code and answer the following subquestions: data(cars) y <- cars$dist x <- cars$speed car_mod <- lm(y ~ x, data = cars) coef(car_mod) ## Output > coef(car_mod) (Intercept) x -17.579095 3.932409 (a) Find the best fit line $\hat{y}$. [2 Marks] (b) Find the residual corresponding to the data point $(x, y) = (4, 2)$. [3 Marks] 2. Let $X_1, X_2, \ldots, X_n$ be i.i.d. samples from a normal distribution with mean $\mu$ and variance $\sigma^2$. Consider the sample mean and sample variance as $$\overline{X} = \frac{1}{n}\sum_{i=1}^{n} X_i, \qquad\qquad S^2 = \frac{1}{(n-1)}\sum_{i=1}^{n}(X_i - \overline{X})^2,$$ then which of the following are true? Select all that apply [3 Marks] (a) $\overline{X} \sim \text{Normal}(\mu, \sigma^2/n)$ (b) $\overline{X} \sim \text{Normal}(\mu, n\sigma^2)$ (c) $\dfrac{(n-1)S^2}{\sigma^2} \sim \chi^2_n$ (d) $\dfrac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t_n$ (e) $\dfrac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t_{(n-1)}$ 3. Consider the first 10 rows of the `iris` dataset as below: head(iris, 10) ## output Sepal.Length Sepal.Width Petal.Length Petal.Width Species 1 5.1 3.5 1.4 0.2 setosa 2 4.9 3.0 1.4 0.2 setosa 3 1.0 1.0 1.0 1.0 <NA> 4 4.6 3.1 1.5 0.2 setosa 5 5.0 3.6 1.4 0.2 setosa 6 5.4 3.9 1.7 0.4 setosa 7 2.0 2.0 2.0 2.0 <NA> 8 5.0 3.4 1.5 0.2 setosa 9 3.0 3.0 3.0 3.0 <NA> 10 4.9 3.1 1.5 0.1 setosa (a) What will be the output of `iris[7, 5]`? [1 Mark] (b) What will be the output of the following code snippet? [2 Marks] iris$Sepal.Length[c(1, 4, 7)] = c(5, 5, 5) iris[c(1, 4, 5), ] (c) The `iris` dataset contains a total of 150 observations. Consider the following output: var(iris$Sepal.Length) [1] 0.8636282 Then, find the population variance of the variable `Sepal.Length`. [2 Marks] 4. Consider the linear model : $$y_{ij} = \alpha + \beta x_{ij} + \gamma_i + \epsilon_{ij} \quad ; \quad 1 \leq j \leq n_i \, , \, 1 \leq i \leq 3,$$ where : - $\gamma_i \sim N(0, \sigma^2_\gamma)$ is a random effect for group $i$, independent of $\epsilon_{ij}$. - $\epsilon_{ij} \sim N(0, \sigma^2_\epsilon)$ are independent errors. - $x_{ij}$ is a known covariate. Find the distribution of $y_{ij}$. [2 Marks] 5. Suppose you fit a linear regression model to the following data: | $\vec{x}$ | $\vec{y}$ | |---|---| | 1 | 2 | | 2 | 4 | | 3 | 5 | | 4 | 4 | | 5 | 5 | and the estimated regression equation is : $\hat{\vec{y}} = 2.2 + 0.6x$. If the *prediction errors* be defined by $\vec{e}_i = \vec{y}_i - \hat{\vec{y}}_i$, then calculate the value of $\sum_{i=1}^{n}(\vec{e}_i)^2$. [4 Marks] 6. Let $(\vec{Y}, X\vec{\beta}, \sigma^2 I_n)$ be a linear model and $p \in \mathbb{R}^m$. Prove that $p^T\vec{\beta}$ is estimable $\Leftrightarrow p \in C(X^T)$. [4 Marks] 7. Consider a model: $$\begin{aligned} y_1 &= 2\beta_1 + 2\beta_2 + 2\beta_3 + 2\beta_4 + \epsilon_1 \,; \\ y_2 &= 2\beta_1 + 2\beta_3 - 2\beta_2 - 2\beta_4 + \epsilon_2 \,; \\ y_3 &= 2\beta_1 + 2\beta_2 - 2\beta_3 - 2\beta_4 + \epsilon_3 \,; \\ y_4 &= 2\beta_1 + 2\beta_4 - 2\beta_2 - 2\beta_3 + \epsilon_4 \,, \end{aligned}$$ where $\beta_i \in R$ and $\epsilon_i$'s are uncorrelated random variables with variance $\sigma^2$; $i = 1, 2, 3, 4$. (a) If we want to rewrite the model as $(\vec{Y}, X\vec{\beta}, I_{4\times4})$, then find $\vec{Y}, X, \vec{\beta}$ and $\vec{\epsilon}$. [2Marks] (b) Define Normal equations and using normal equations find the least square estimates of $\beta_1, \beta_2, \beta_3$ and $\beta_4$. [5 Marks]

Question 1

+30−0Single correct

A linear regression model $\hat{y} = a_0 + a_1 x$ is fit for the cars dataset. The dataset has 50 observations with 2 variables dist and speed. Study the code and answer the following subquestions:

1data(cars)
2y <- cars$dist
3x <- cars$speed
4
5car_mod <- lm(y ~ x, data = cars)
6coef(car_mod)
7
8## Output
9> coef(car_mod)
10(Intercept)           x
11 -17.579095    3.932409

(a) Find the best fit line $\hat{y}$ . [2 Marks]

(b) Find the residual corresponding to the data point $(x, y) = (4, 2)$ . [3 Marks]

Let $X_1, X_2, \ldots, X_n$ be i.i.d. samples from a normal distribution with mean $\mu$ and variance $\sigma^2$ . Consider the sample mean and sample variance as

$\overline{X} = \frac{1}{n}\sum_{i=1}^{n} X_i, \qquad\qquad S^2 = \frac{1}{(n-1)}\sum_{i=1}^{n}(X_i - \overline{X})^2,$

then which of the following are true? Select all that apply [3 Marks]

(a) $\overline{X} \sim \text{Normal}(\mu, \sigma^2/n)$

(b) $\overline{X} \sim \text{Normal}(\mu, n\sigma^2)$

(d) $\dfrac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t_n$

(e) $\dfrac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t_{(n-1)}$

Consider the first 10 rows of the iris dataset as below:

1head(iris, 10)
2
3## output
4   Sepal.Length Sepal.Width Petal.Length Petal.Width Species
51           5.1         3.5          1.4         0.2  setosa
62           4.9         3.0          1.4         0.2  setosa
73           1.0         1.0          1.0         1.0    <NA>
84           4.6         3.1          1.5         0.2  setosa
95           5.0         3.6          1.4         0.2  setosa
106           5.4         3.9          1.7         0.4  setosa
117           2.0         2.0          2.0         2.0    <NA>
128           5.0         3.4          1.5         0.2  setosa
139           3.0         3.0          3.0         3.0    <NA>
1410          4.9         3.1          1.5         0.1  setosa

(a) What will be the output of iris[7, 5]? [1 Mark]

(b) What will be the output of the following code snippet? [2 Marks]

1iris$Sepal.Length[c(1, 4, 7)] = c(5, 5, 5)
2iris[c(1, 4, 5), ]

1var(iris$Sepal.Length)
2[1] 0.8636282

Then, find the population variance of the variable Sepal.Length. [2 Marks]

Consider the linear model :

$y_{ij} = \alpha + \beta x_{ij} + \gamma_i + \epsilon_{ij} \quad ; \quad 1 \leq j \leq n_i \, , \, 1 \leq i \leq 3,$

where :

$\gamma_i \sim N(0, \sigma^2_\gamma)$ is a random effect for group $i$ , independent of $\epsilon_{ij}$ .
$\epsilon_{ij} \sim N(0, \sigma^2_\epsilon)$ are independent errors.
$x_{ij}$ is a known covariate.

Find the distribution of $y_{ij}$ . [2 Marks]

Suppose you fit a linear regression model to the following data:

$\vec{x}$	$\vec{y}$
1	2
2	4
3	5
4	4
5	5

Let $(\vec{Y}, X\vec{\beta}, \sigma^2 I_n)$ be a linear model and $p \in \mathbb{R}^m$ . Prove that $p^T\vec{\beta}$ is estimable $\Leftrightarrow p \in C(X^T)$ . [4 Marks]
Consider a model:

where $\beta_i \in R$ and $\epsilon_i$ 's are uncorrelated random variables with variance $\sigma^2$ ; $i = 1, 2, 3, 4$ .

(a) If we want to rewrite the model as $(\vec{Y}, X\vec{\beta}, I_{4\times4})$ , then find $\vec{Y}, X, \vec{\beta}$ and $\vec{\epsilon}$ . [2Marks]

(b) Define Normal equations and using normal equations find the least square estimates of $\beta_1, \beta_2, \beta_3$ and $\beta_4$ . [5 Marks]

A.
I have written answers on the answer sheets
B.
Not applicable

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