Linear Regression Models and Fit - Regression Analysis - Statistics Previous Year Questions

1

2019 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2019

Let \(Y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon_i\) for \(i = 1, ..., 10\), where \(x_{1i}\)'s and \(x_{2i}\)'s are fixed covariates, and \(\epsilon_i\)'s are uncorrelated random variables with mean 0 and unknown variance \(\sigma^2\). Here \(\beta_0, \beta_1\) and \(\beta_2\) are unknown parameters. Further, define \(\hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_{1i} + \hat{\beta}_2 x_{2i}\), where \((\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2)\) is the unbiased least squares estimator of \((\beta_0, \beta_1, \beta_2)\). Then an unbiased estimator of \(\sigma^2\) is

A

\(\dfrac{\sum_{i=1}^{10}(Y_i - \hat{Y}_i)^2}{10}\)

B

\(\dfrac{\sum_{i=1}^{10}(Y_i - \hat{Y}_i)^2}{9}\)

C

\(\dfrac{\sum_{i=1}^{10}(Y_i - \hat{Y}_i)^2}{8}\)

D

\(\dfrac{\sum_{i=1}^{10}(Y_i - \hat{Y}_i)^2}{7}\)

Open complete paper

2

2019 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2019

For \(i = 1, 2, 3\), let \(Y_i = \alpha + \beta x_i + \epsilon_i\), where \(x_i\)'s are fixed covariates, and \(\epsilon_i\)'s are independent and identically distributed standard normal random variables. Here, \(\alpha\) and \(\beta\) are unknown parameters. Given the following observations, the best linear unbiased estimate of \(\alpha + \beta\) is equal to

A

1.5

B

1

C

1.8

D

2.1

Open complete paper

3

2019 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2019

Consider a fixed effects two-way analysis of variance model \(Y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + \epsilon_{ijk}\), where \(i = 1, ..., a; j = 1, ..., b; k = 1, ..., r\), and \(\epsilon_{ijk}\)'s are independent and identically distributed normal random variables with zero mean and constant variance. Then the degrees of freedom available to estimate the error variance is zero when

A

a = 1

B

b = 1

C

r = 1

D

None of the above.

Open complete paper

4

2019 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2019

For \(i = 1,2,3,4\), let \(Y_i = \alpha + \beta x_i + \epsilon_i\) where \(x_i\)'s are fixed covariates and \(\epsilon_i\)'s are uncorrelated random variables with mean 0 and variance 3. Here, \(\alpha\) and \(\beta\) are unknown parameters. Given the following observations,

\(Y_i\)	2	2.5	-0.5	1
\(x_i\)	3	2	-4	-1

the variance of the least squares estimator of \(\beta\) is equal to ...

Check answer

Open complete paper

5

2019 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2019

Let \( Y_i = \alpha + \beta x_i + \epsilon_i \), where \( i=1,2,3,4 \), \( x_i \)'s are fixed covariates and \( \epsilon_i \)'s are independent and identically distributed standard normal random variables. Here, \( \alpha \) and \( \beta \) are unknown parameters. Let \( \Phi \) be the cumulative distribution function of the standard normal distribution and \( \Phi(1.96) = 0.975 \). Given the following observations,

\( Y_i \)	3	-2.5	5	-5
\( x_i \)	1	-2	3	-2

the length (rounded off to two decimal places) of the shortest 95% confidence interval for \( \beta \) based on its least squares estimator is equal to ...

Check answer

Open complete paper

6

2019 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2019

Let \(Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \; i = 1, ..., n\), where \(x_i\)'s are fixed covariates, and \(\epsilon_i\)'s are uncorrelated random variables with mean zero and constant variance. Suppose that \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the least squares estimators of the unknown parameters \(\beta_0\) and \(\beta_1\), respectively. If \(\sum_{i=1}^n x_i = 0\), then the correlation between \(\hat{\beta}_0\) and \(\hat{\beta}_1\) is equal to ...

Check answer

Open complete paper

7

2020 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2020

Consider the regression model \(Y_i = \beta_0 + \beta_1 x_i^2 + \epsilon_i, \ i = 1, 2, \ldots, n \ (n \geq 2) ;\) where \(\beta_0\) and \(\beta_1\) are unknown parameters and \(\epsilon_i\)'s are random errors. Let \(y_i\) be the observed value of \(Y_i\), \(i = 1, \ldots, n\). Using the method of ordinary least squares, the estimate of \(\beta_1\) is

A

\(\frac{1}{n} \left( \frac{n \sum x_i^2 y_i - (\sum y_i)(\sum x_i^2)}{\sum x_i^4 - (\sum x_i^2)^2} \right)\)

B

\(\frac{n \sum x_i^2 y_i - (\sum y_i)(\sum x_i^2)}{n \sum x_i^4 - (\sum x_i^2)^2}\)

C

\(\frac{\sum x_i^2 y_i - (\sum y_i)(\sum x_i^2)}{\sum x_i^4 - (\sum x_i^2)^2}\)

D

\(\frac{\sum x_i^2 y_i - n(\sum y_i)(\sum x_i^2)}{\sum x_i^4 - n(\sum x_i^2)^2}\)

Open complete paper

8

2020 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2020

Consider a two-way fixed effects analysis of variance model without interaction effect and one observation per cell. If there are 5 factors and 4 columns, then the degrees of freedom for the error sum of squares is

A

\(20\)

B

\(19\)

C

\(12\)

D

\(11\)

Open complete paper

9

2020 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2020

Let \( Y_i = \alpha + \beta x_i + \epsilon_i, \; i = 1, 2, ..., 7 \), where \( x_i \)'s are fixed covariates and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance. Suppose that \( \hat{\alpha} \) and \( \hat{\beta} \) are the least squares estimators of \( \alpha \) and \( \beta \), respectively. Given the following data:
\[ \sum_{i=1}^7 x_i = 0, \quad \sum_{i=1}^7 x_i^2 = 28, \quad \sum_{i=1}^7 x_i y_i = 28, \quad \sum_{i=1}^7 y_i = 21 \text{ and } \sum_{i=1}^7 y_i^2 = 91, \]
where \( y_i \) is the observed value of \( Y_i, \; i = 1, ..., 7 \). Then the correlation coefficient between \( \hat{\alpha} \) and \( \hat{\beta} \) equals ___________

Check answer

Open complete paper

10

2020 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2020

A simple linear regression model with unknown intercept and unknown slope is fitted to the following data

\(x\)	-2	-1	0	1	2
\(y\)	3	5	8	9	10

using the method of ordinary least squares. Then the predicted value of \(y\) corresponding to \(x = 5\) is ____________________

Check answer

Open complete paper

11

2021 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2021

Consider the simple linear regression model \( Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \; i = 1, 2, ..., n \; (n \geq 3) \), where \( \beta_0 \) and \( \beta_1 \) are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance \( \sigma^2 > 0 \). Suppose that \( \hat{\beta}_0 \) and \( \hat{\beta}_1 \) are the ordinary least squares estimators of \( \beta_0 \) and \( \beta_1 \), respectively. Define \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \), \( S_1 = \sum_{i=1}^n (x_i - \bar{x})^2 \) and \( S_2 = \sum_{i=1}^n y_i (x_i - \bar{x}) \), where \( y_i \) is the observed value of \( Y_i \), \( i = 1, 2, ..., n \). Then for a real constant \( c \), the variance of \( \hat{\beta}_0 + c \hat{\beta}_1 \) is

A

\(\sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_2} \right) \)

B

\(\sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_1} \right) \)

C

\(\frac{\sigma^2}{n} \)

D

\(\sigma^2 \left( \frac{1}{n} + \frac{\bar{x}^2}{S_2} \right) + c^2 \)

Open complete paper

12

2021 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2021

Let \(Y_t = \alpha + \beta x_t + \epsilon_t\), \(t = 1,2,3\), where \(x_t\)'s are fixed covariates, \(\alpha\) and \(\beta\) are unknown parameters and \(\epsilon_t\)'s are independent and identically distributed random variables with mean zero and finite variance. Let \(\hat{\alpha}\) and \(\hat{\beta}\) be the ordinary least squares estimators of \(\alpha\) and \(\beta\), respectively. Given the following observations

\(y_t\)	8.62	26.86	54.02
\(x_t\)	3.29	21.53	48.69

the value of \(\hat{\alpha} + \hat{\beta}\) equals __________ (round off to 2 decimal places).

Check answer

Open complete paper

13

2021 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2021

Consider the multiple linear regression model
\( Y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + ... + \beta_{22} x_{22,i} + \epsilon_i, \;\; i = 1, 2, ..., 123, \)
where, for \( i = 0, 1, 2, ..., 22 \), \( \beta_j \)'s are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed \( N(0, \sigma^2) \), \( \sigma > 0 \), random variables.
If the sum of squares due to regression is 338.92, the total sum of squares is 522.30 and \( R^2_{adj} \) denotes the value of adjusted \( R^2 \), then \( 100 R^2_{adj} \) equals ________ (round off to 2 decimal places).

Check answer

Open complete paper

14

2022 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2022

Consider the multiple regression model
\( Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon, \)
where \( \epsilon \) is normally distributed with mean 0 and variance \( \sigma^2 > 0, \) and \( \beta_0, \beta_1, \beta_2, \beta_3 \) are unknown parameters. Suppose 52 observations of \( (Y, X_1, X_2, X_3) \) yield sum of squares due to regression as 18.6 and total sum of squares as 79.23. Then, for testing the null hypothesis \( H_0: \beta_1 = \beta_2 = \beta_3 = 0 \) against the alternative hypothesis \( H_1: \beta_i \neq 0 \) for some \( i = 1, 2, 3, \) the value of the test statistic (rounded off to three decimal places), based on one way analysis of variance, is ______

Check answer

Open complete paper

15

2022 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2022

Let a linear model \(Y = \beta_0 + \beta_1 X + \epsilon\) be fitted to the following data, where \(\epsilon\) is normally distributed with mean 0 and unknown variance \(\sigma^2 > 0\).

\(x_i\)	0	1	2	3	4
\(y_i\)	3	4	5	6	7

Let \(\hat{Y}_0\) denote the ordinary least-square estimator of \(Y\) at \(X = 6\), and the variance of \(\hat{Y}_0 = c\sigma^2\). Then the value of the real constant \(c\) (rounded off to one decimal place) is equal to ______

Check answer

Open complete paper

16

2023 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2023

Consider the following regression model
\[ y_k = \alpha_0 + \alpha_1 \log_e k + \epsilon_k, \quad k = 1, 2, \ldots, n, \]
where \( \epsilon_k \)'s are independent and identically distributed random variables each having probability density function \( f(x) = \frac{1}{2} e^{-|x|}, \; x \in \mathbb{R} \). Then which one of the following statements is true?

A

The maximum likelihood estimator of \(\alpha_0 \) does not exist

B

The maximum likelihood estimator of \(\alpha_1 \) does not exist

C

The least squares estimator of \(\alpha_0 \) exists and is unique

D

The least squares estimator of \(\alpha_1 \) exists, but it is not unique

Open complete paper

17

2023 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2023

Consider the following regression model \( y_t = a_0 + a_1 t + a_2 t^2 + \epsilon_t, \quad t = 1, 2, ..., 100, \) where \( a_0, a_1 \) and \( a_2 \) are unknown parameters and \( \epsilon_t \)’s are independent and identically distributed random variables each having \( N(\mu, 1) \) distribution with \( \mu \in \mathbb{R} \) unknown. Then which of the following statements is/are true?

A

There exists an unbiased estimator of \(a_1 \)

B

There exists an unbiased estimator of \(a_2 \)

C

There exists an unbiased estimator of \(a_0 \)

D

There exists an unbiased estimator of \(\mu \)

Check answer

Open complete paper

18

2024 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2024

Consider the simple linear regression model \(y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad i = 1, 2, \ldots, n,\) where \(\beta_0\) and \(\beta_1\) are unknown parameters, \(\varepsilon_i\)'s are uncorrelated random errors with mean 0 and finite variance \(\sigma^2 > 0\). Let \(\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\) and \(\widehat{\beta_1}\) be the least squares estimator of \(\beta_1\). Then which one of the following statements is true?

A

The covariance between \(\bar{y}\) and \(\widehat{\beta_1}\) is less than 0

B

The covariance between \(\bar{y}\) and \(\widehat{\beta_1}\) is greater than 0

C

The covariance between \(\bar{y}\) and \(\widehat{\beta_1}\) is equal to 0

D

The covariance between \(\bar{y}\) and \(\widehat{\beta_1}\) does not exist

Open complete paper

19

2024 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2024

Consider the simple linear regression model
\[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad i = 1, 2, \ldots, n \quad (n \geq 3), \]
where \(\beta_0\) and \(\beta_1\) are unknown parameters, \(\varepsilon_i\)'s are uncorrelated random errors with mean \(0\) and finite variance \(\sigma^2 > 0\). Let \(\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\) and \(\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i\), \(i = 1, 2, \ldots, n\), where \(\hat{\beta}_0\) and \(\hat{\beta}_1\) represent least squares estimators of \(\beta_0\) and \(\beta_1\), respectively. Let \(T_1 = \frac{1}{n-2} \sum_{i=1}^n (y_i - \hat{y}_i)^2\) and \(T_2 = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2\). Then which one of the following statements is true?

A

Both \(T_1\) and \(T_2\) are unbiased estimators of \(\sigma^2\)

B

\(T_1\) is an unbiased estimator of \(\sigma^2\), but \(T_2\) is not an unbiased estimator of \(\sigma^2\)

C

\(T_1\) is not an unbiased estimator of \(\sigma^2\), but \(T_2\) is an unbiased estimator of \(\sigma^2\)

D

Neither \(T_1\) nor \(T_2\) is an unbiased estimator of \(\sigma^2\)

Open complete paper

20

2024 · Statistics · Regression Analysis · Linear Regression Models and Fit

Statistics (ST) 2024

Consider the linear regression model
\( y_i = \beta_1 x_i + \epsilon_i, \; i = 1, 2, ..., n, \)
where \( \beta_1 \) is an unknown parameter, \( \epsilon_i \)'s are uncorrelated random errors with mean 0 and finite variance \( \sigma^2 > 0 \). The five data points \( (x_1, y_1) = (2, 5), (x_2, y_2) = (1, 6), (x_3, y_3) = (3, 4), (x_4, y_4) = (2, 3) \) and \( (x_5, y_5) = (4, 6) \) yield the least squares estimate of \( \beta_1 \) to be equal to __________ (rounded off to two decimal places).

Check answer

Open complete paper