Solve your IGNOU Doubts

Question:

A company conducted a study on its employees to see the relationship of several variables with an employee’s IQ. For this purpose, fifteen employees were selected, and an IQ test and five different personality tests were given to them. Each employee’s IQ was recorded along with scores on five tests. The data are shown in the following table

Employee	Test 1	Test 1	Test 3	Test 4	Test 5	IQ
1	83	80	78	77	67	99
2	73	85	67	80	63	92
3	81	80	71	81	68	94
4	96	86	82	83	56	99
5	84	73	75	75	68	94
6	72	74	71	67	59	79
7	84	79	84	84	69	97
8	54	86	61	69	53	92
9	86	85	79	78	76	94
10	42	71	60	80	56	86
11	83	72	72	78	74	98
12	63	86	65	85	56	83
13	69	76	64	85	61	98
14	81	84	65	85	64	96

Determine the most appropriate regression model for the employee’s IQ using a stepwise approach at a 5 % level of significance and interpret the results. Does the final regression model satisfy the linearity and normality assumptions?

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Question:

For the data given in Question 2(b), obtain the followings:

(i) Diagonal of the hat matrix. Also, check the leverage points, if any.

(ii) Cook’s Distances. Also, verify the influence points, if any.

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Question:

Explain the assumptions underlying the multiple linear regression model.

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Question:

Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in mg/dL) and systolic blood pressure (in mm/Hg) were obtained for 15 patients to explore the relationship between cholesterol and blood pressure:

S. No	. Cholesterol (mg/dL)	SBP (mm/Hg)
1	300	150
2	410	270
3	380	210
4	530	310
5	570	350
6	490	310
7	340	210
8	320	150
9	280	110
10	550	320
11	340	220
12	350	170
13	410	260
14	390	230
15	450	270

(i) Fit a linear regression model using the method of least squares.

(ii) Construct the normal probability plot for the regression model fitted on serum cholesterol and systolic blood pressure.

(iii) Test the significance of the fitted regression model.

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Question:

Write a short note on the problem of multicollinearity and autocorrelation.

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Question:

State whether the following statements are true or false and also give the reason in support of your answer.

(i) We define three indicator variables for an explanatory variable with three categories.

(ii) If the coefficient of determination is 0.833, the number of observations and explanatory variables are 12 and 3, respectively, then the Adjusted R² will be 0.84.

(iii) For a simple regression model fitted on 15 observations, if we have h₁ = 0.37, then it is an indication to trace the leverage point in the regression model.

(iv) In a regression model $Y=\beta_0+\beta_1 X_1+\beta_2 X_2+\varepsilon,\text{if}H_0:\beta_1=0$ is not rejected, then the variable X₁ will remain in the model.

(v) The logit link function is log [-log(1-π)].

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Question:

$\text{If}\underline{X}\sim N_3(\underline{\mu},\underline{\Sigma})\text{with}\underline{\mu}=\begin{pmatrix}2\\1\\2\end{pmatrix}\text{and}\underline{\Sigma}=\begin{pmatrix}5&3&0\\3&3&-2\\0&-2&5\end{pmatrix}.$ Then find the joint distribution of $X_1+2X_2,\quad 2X_1-X_2\quad\text{and}\quad X_3.$

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Question:

Relationship between (ii) and (iii).

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Question:

Clustering

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Question:

Hotelling’s T2

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Question:

Mahalanobis D2

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Question:

Covariance Matrix

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Question:

Obtain the maximum likelihood estimator of the mean vector and variance-covariance matrix of the multivariate normal distribution.

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Question:

Consider the following data of 11 samples on 8 variables by Anscombe, Francis J. (1973):

x₁	x₂	x₃	x₄	y₁	y₂	y₃	y₄
10	10	10	8	8.04	9.14	7.46	6.58
8	8	8	8	6.95	8.14	6.77	5.76
13	13	13	8	7.58	8.74	12.74	7.71
9	9	9	8	8.81	8.77	7.11	8.84
11	11	11	8	8.33	9.26	7.81	8.47
14	14	14	8	9.96	8.10	8.84	7.04
6	6	6	8	7.24	6.13	6.08	5.25
4	4	4	19	4.26	3.10	5.39	12.50
12	12	12	8	10.84	9.13	8.15	5.56
7	7	7	8	4.82	7.26	6.42	5.56
5	7	5	8	5.68	4.74	5.73	5.56

If the vector $\underline{x}=\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}\quad\text{and}\quad\underline{y}=\begin{pmatrix}y_1\\y_2\\y_3\\y_4\end{pmatrix}$ then obtain the sample covariance matrix between $\underline{x}\text{and}\underline{y}$

Source: Anscombe, Francis J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17– 21. doi: 10.2307/2682899.

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Question:

) $\underline{X}\sim N_4(\underline{\mu},\underline{\Sigma}),\quad\text{where}\quad\underline{\mu}=\begin{pmatrix}3\\-2\\1\\-2\end{pmatrix}$ and $\underline{\Sigma}=\begin{pmatrix}4&0&0&0\\0&3&0&0\\0&0&2&-2\\0&0&-2&5\end{pmatrix}$ Check the

independence $\text{of the(i)}X_2\text{and}X_1\quad\text{(ii)}(X_2,X_4)\text{and}(X_1,X_3)\quad\text{(iii)}(X_1,X_2)\text{and}(X_3,X_4)$

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Question:

Let X  be a 3-dimensional random vector with dispersion matrix

$\underline{\Sigma}=\begin{pmatrix}4&-2&0\\-2&4&0\\0&0&2\end{pmatrix}$

Determine the first principal component and the proportion of the total variability that it explains.

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Question:

) $\underline{X}=\begin{pmatrix}\underline{X}^{(1)}\\\underline{X}^{(2)}\end{pmatrix}\sim N_4(\underline{\mu},\underline{\Sigma})$ whare $\underline{\mu}=\begin{pmatrix}-4\\1\\4\\0\end{pmatrix},\quad\underline{\Sigma}=\begin{pmatrix}2&2&|&3&0\\2&1&|&2&0\\--&--&--&--&--\\3&2&|&2&1\\0&0&|&1&1\end{pmatrix}$ Find the

$E(\underline{X}^{(2)}|\underline{X}^{(1)}=\underline{x}^{(1)})\quad\text{and}\quad Cov(\underline{X}^{(2)}|\underline{X}^{(1)}=\underline{x}^{(1)})$

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Question:

$\text{Let}\underline{X}=\begin{pmatrix}X_1\\X_2\end{pmatrix}$ has the following joint density function

$f(x_1,x_2)=\begin{cases}4x_1x_2&,0<x_1<1,0<x_2<1\\0&,\text{otherwise}.\end{cases}$

Find the marginal distributions, mean vector and variance-covariance matrix. Also, comment on the independence of X₁and X₂

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Question:

State whether the following statements are true or false and also give the reason in support of your answer:

(a) The covariance matrix of random vectors $\underline{X}\text{and}\underline{Y}$ is symmetric.

(b) $if\underline{x}$ is a p-variate normal random vector, then every linear combination $\underline{c{}'}$ $\underline{x}$ is a scalar vector, is also p-variate normal vector

(d) If a matrix is positive definite then its inverse is also positive definite.

(e) $\text{If}\underline{X}\sim N_2\left(\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right)\text{and}\underline{Y}\sim N_2\left(\begin{pmatrix}-1\\3\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right),\text{then}$

$\underline{X}+\underline{Y}\sim N_2\left(\begin{pmatrix}1\\4\end{pmatrix},\begin{pmatrix}2&0\\0&2\end{pmatrix}\right).$

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Question:

Define the Residual B.I.B.D. and Derived B.I.B.D. of a given symmetric B.I.B.D. Mention the rule of constructing a residual B.I.B.D. corresponding to a specific B.I.B.D.

$\text{Let}\mathbf{X}=\{1,2,3,4,5\}\text{and}\mathcal{A}=\{\{1,2,3\},\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,3,5\},\{1,4,5\},\dots\}$