Solve your IGNOU Doubts

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 $\underset{\sim }{X}$ and $\underset{\sim }{Y}$ is symmetric.

(b) If $\underset{\sim }{X}$ is a p-variate normal random vector, then every linear combination $\underset{\sim }{C{}'}\underset{\sim }{X},$ where $\underset{\sim }{C}_{P\times 1}$ is a scalar vector, is also p-variate normal vector.

(c) The trace of matrix $\begin{pmatrix} 3 & -2\\ -2 &6 \end{pmatrix}$ is 9.

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

(e) If $\underset{\sim }{X}$ $-N_2$ $\left ( \binom{2}{1},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right )$ and

$\underset{\sim }{Y}$ $-N_2$ $\left ( \binom{-1}{3},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right ),$ then

$\underset{\sim }{X}$ $+$ $\underset{\sim }{Y}$ $-N_2$ $\left ( \binom{1}{1},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right )$

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Question:

(i) Curve of F-distribution
(ii) Mean Squared Error

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Question:

Complete the following table, one is done for you:

S.
No
.

Test For

Name of the Test

Test Statistic

1

Population mean when population variance is known and population is normal

Z-test

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	$Z=\frac{\bar{X}-\mu}{\sigma /\sqrt{n}}$

2

Population mean when population variance is unknown and population is normal

3

Difference of two population means when samples are paired, and population of differences follows normal distribution.

4

Difference of two population means when samples are independent, and population of differences follows normal distribution.

5

Population variance when the population is normal distributed

6

Population variance when the population is not normal distributed

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Question:

An engineer conducted an experiment to compare two metals: iron and copper, as bonding agents for an alloy material. Components of the alloy were bonded using the metals as bonding agents, and the pressures required to break the bonds were measured. The data for the breaking pressures are given in the following table:

S. No.	Breaking Pressure
S. No.	Iron	Copper
1	72.7	73
2	69.6	67.2
3	83.4	75.3
4	78.9	61.4
5	75	74
6	71.6	69.5
7	85.7	69.8
8	73.5	73.8
9	70.4	68
10	84.2	76.1

If the breaking pressures for both iron and copper are normally distributed, are the variances of the distributions of the breaking pressure of iron and copper equal at 5 % level of significance?

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Question:

A sample of 500 shops was selected in a large metropolitan area to determine various information concerning consumer behaviour. One question, among the questions, asked, was “Do you enjoy shopping for clothing?” Out of 240 males 136 answered yes. Out of 260 females, 224 answered yes. Find 95% confidence interval for the difference of the proportions for enjoys shopping for clothing.

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Question:

The department of transportation has mandated that the average speed of cars on interstate
highways be no more than 70 km per hours in order. To check that the people follow it or not,
a researcher took a random sample of 186 cars and found that the average speed was 72 km
per hours with a standard deviation 0.6 km per hours.
(a) Construct the interval around the sample mean that would contain the population mean 95% of the time.
(b) If the researcher wants to test that the true mean speed on its highways is 70 km per hours or less with 95% confidence then
(i) State null and alternative hypotheses.
(ii) Name the test which is suitable in this situation and why?
(iii) Calculate the value of test statistic and critical value.
(iv)Draw the conclusion on the basis of the applied test

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Question:

A baby-sister has 6 children under her supervision. The age of each child is as follows:

Child	Age (in years)
Sonu	2
Lavnik	4
Chiya	3
Amam	3
Avishi	4
Ridhi	5
Sidhi	3

(i) What is the form of population of age of children?
(ii) Prepare the sampling distribution of sample mean when sample size is 2.
(iii) Is the shape of the sampling distribution normal?
(iv) Calculate the mean and standard error of the sampling distribution.

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Question:

Describe the various forms of the sampling distribution of ratio of two sample variances.

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Question:

State whether the following statements are True or False. Give reason in support of your answer:

(i) if $X_1,X_2,X_3,X_4$ and $X_5$ is a random sample of size 5 taken from an Exponential distribution, then estimator T1 is more efficient than $T_2$ . $T_1=\frac{X_1+X_2+X_3+X_4+X_5}{5},\, T_2=\frac{X_1+2X_2+3X_3+4X_4+5X_5}{15}$

(ii) if $T_1$ and $T_2$ are two estimators of the parameter $\theta$ such thet Var $\left (T_1 \right )=1/n$ and Var $\left (T_2 \right )$ then $T_1$ is more efficient then $T_2.$

(iii) A 95% confidence interval is smaller than 99% confidence interval.

(iv) If the probability density function of a random variable X follows F-distribution is

$f(x)=\frac{1}{(1+x)},x\geq 0$

then degrees of freedom of the distribution will be (2,2).

(v) A patient suffering from fever reaches to a doctor and suppose the doctor formulate the hypotheses as

$H_0:$ The patient is a chikunguniya patient

$H_1:$ The patient is not a chikunguniya patient

If the doctor rejects H0 when the patient is actually a chikunguniya patient, then the doctor commits type II error.

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Question:

The following data on diagnosis of coronary heart disease (where 0 indicating absence and 1 indicating presence), serum cholesterol (in mg/dl), resting blood pressure (in mmHg) and weight (in kg) were obtained for 80 patients to explore the relationship of coronary heart disease with cholesterol and weight:

S. No.	Serum Cholesterol (mg/dl)	Weight (kg)	Number of Patients having CHD	Total Number of Patients
1	420	60	10	20
2	450	68	15	30
3	400	54	4	15
4	510	74	2	10
5	480	62	1	5

(i) Fit a multiple logistic model for the dependence of coronary heart disease on the average serum cholesterol and weight considering $\hat{\beta}_{0}^{0}=1.279,\, \hat{\beta}_{1}^{0}=-0.035$ and $\hat{\beta}_{2}^{0}=0.172$ as the initial vales of th parameters (solve only for one lteration ).

(ii) Test the significance of the fitted model using Hosmer-Lemeshow test at 5% level of significance.

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Question:

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

Employee	Test 1	Test 2	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	79	64	96
15	50	85	71	65	75	76

Determine the most appropriate regression model for the employee’s IQ using stepwise approach at 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 and also check the leverage points if any.
(ii) Cook’s Distances, DFFITS and DFBETAS. Also verify the influence points if any

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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:

Explain the assumptions underlying multiple linear 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 $h_{ii}=0.37$ , then it is an indication to trace the leverage point in the regression model.

(iv) In a regression model $Y=\beta _0+\beta_1X_1+\beta_2X_2+\varepsilon ,$ 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 $\left [ -log(1-\pi) \right ].$

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Question:

Explain in detail the basic criteria for selecting content including media in the development of curriculum.

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Question:

Curriculum is plan of action". Discuss thestatement with suitable examples in thecontext of distance education.

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Question:

Explain the concept of formative and summative evaluation in distance education

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Question:

Explain the criteria used to evaluate a programme offered by an open and distance education Institution, with examples.

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