Solve your IGNOU Doubts

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 data 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:

A random sample of 250 patients was selected and their workout timing and diabetes status were recorded. The following table shows the workout timing and severity of diabetes:

Workout (in minutes)	Severity of diabetes status
Workout (in minutes)	Low	Moderate	High
0 −15	06	27	19
15 to 30	08	38	17
30 to 45	21	45	33
≥ 45	14	18	06

Test at 5% level of significance whether workout habit and diabetes are associated with to each other or not.

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Question:

Differentiate between Chi-square tests for association and homogeneity of proportions. Also mention the assumptions of these tests.

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Question:

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

(a) The value of sensitivity of the following results of a diagnostic test is 0.85.

Disease	Test result		Total
Disease	+	-	Total
Present	170	30	200
Absent	20	280	300

(b) For the following cohort study, the relative risk for the lung cancer among smokers is 3.5.

	Lung Cancer	No Lung Cancer	Totel
Smokers	100	1220	1320
Non-smokers	50	2260	2310

(c) The logit link function is log $\left [ -log(1-\pi ) \right ],$
(d) We define three indicator/dummy variables for a regressor variable with three categories.
(e) Left censoring occurs whenever the exact time of occurrence of an event is not known.

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Question:

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Question:

a) The population dynamics of a species is governed by the discrete model $x_{a+1=x_{a}}exp\left [ r\left ( 1-\frac{x_{n}}{K} \right ) \right ],$ where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population fo $r=2+\varepsilon ,$ where $0< \varepsilon < < 1.$ Since a species becomes extinct if $x_{n\leq 1} n> 1,$ $x_{n\leq 1}$ for any $n> 1,$ show using iterations, that irrespective of the size of $r > 1$ the species could become extinct if the carrying capacity $k <$ r exp $\left [ 1+e^{r-1} -2r\right ].$

b) Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.

$\frac{dn_{t}}{dt}=r_{0N_{1}}-r_{1C_{O}}N_{1}-bN_{1}N_{2}$

$\frac{dn_{2}}{dt}=d_{0}N_{2}-d_{1}V_{0}N_{2+}\beta _{0}bN_{1}N_{2}$

$\frac{dC_{0}}{dt}=k_{1}P-g_{1}C_{0}-m_{1}C_{0}$ $\frac{dV_{0}}{dt}=k_{2}P-g_{2}V_{0}-m_{2}V_{0}$

$\frac{dP}{dt}=Q-hP-kP(N_{1}+N_{2})+gC_{0}N_{1}+lV_{0}N_{2 .}$

Where all the variables and constants are same as defined in the system (32)-(35) except for the following

Q = constant input rate h = decay rate )t

(P = environmental toxicant concentration

k = ingestion rate of toxicant by the populations

$g,l=$ return rate of toxicant in the environment after the death of the populations, assuming that toxicant is non-degradable

$Q> 0,h,k,g,l$ are positive constants.

Do the stability analysis of the following competing species system of equations with diffusion and advection

$\frac{\partial N_{1}}{\partial t}=a_{1}N_{1}-b_{1}N_{1}N_{1}+D_{1}\frac{\partial ^{2}N_{1}}{\partial X^{2}}-V_{1}\frac{\partial N_{1}}{\partial X}$

$\frac{\partial N_{2}}{\partial t}=-d_{1}N_{2}+C_{1}N_{1}N_{2}+D_{2}\frac{\partial ^{2}N_{2}}{\partial X}, 0\leq X\leq L$

where $V_{1}$ and $V_{2}$ are advection velocities in x direction of the two populations with densities $N_{1}$ and $N_{2}$ respectively. $a_{1}$ is the growth rate, $b_{1}$ is the predation rate $d_{1}$ is the death rate $c_{1}$ $C_{1}$ is the conversion rate. $D_{1}$ and $D_{2}$ are diffusion coefficients. The initial and boundary conditions are:

$N_i$ $N_i(X,0))=f_i(X)> 0,0\leq X\leq L,i=1,2.$

$N_i=\bar{N_{1}}$ at $X=0L\forall t,i=2.$

where $N_i$ are the equilibrium solutions of the given system of equations.

Interpret the solution obtained and also write the limitations of the model.

Maximize $Z=3X+_{1+X_{2}}+3X_{3,}$ subject to the constraints

$-X_{1}+2X_{2}+X_{3}\leq 4,4X_{2}-3X_{2}+2X_{3}\leq 3$ and $(X_{1,}X_{2,}X_{3})\geq 0$ and are integers.

b) Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of interarrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?

c) A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 1 every 6 minutes and service time follows exponential distribution with a mean of 10 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.

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Question:

Consider a discrete model given by

$N_{t+1}=\frac{rN_{t}}{1+bN^{2}_{t-1}}=f(N_{t})r> 1.$

Investigate the linear stability about the positive steady state $N^{*}$ by setting

$N_{t=}N^{*}+n_{t.}$ Show that $n_{t}$ satisfies the equation

$n_{t+1-n_{t+}}2(r-1)r^{-}n_{t-1}=0.$

Hence show that r = 2 is a bifurcation value and that as r → 2 the steady state bifurcates to a periodic solution of period 6.

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Question:

Suppose you try a regimen A on 1000 subjects and regimen B on 1600 subjects. Results of the trial show that efficacies of regimen A and B are 76% and 82% respectively. Suppose doctor determines 4% as superiority margin. Can you consider regimen B as superior to regimen A

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Question:

Creatinine excretion is a parameter of kidney function. Generally speaking, lower values indicate better health. This depends on body weight. A researcher conducted a study on creatinine excretion in test group and control group to find the efficacy of a new drug. The subjects were randomly divided. He included 100 subjects in each group but for this exercise consider only 10 subjects in each group. The data obtained on creatinine level in these 10 subjects are as follows:

Test group: 16.6 19.8 17.1 17.0 15.6 20.3 24.7 18.5 17.6 22.0

Control group: 23.2 22.0 21.9 16.7 14.2 23.2 24.8 25.5 28.1 21.8

Do you think that creatinine excretion was really lower in the test group on average?

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Question:

The population consisting of all married couples is collected. The data showing the age of 12 married couples is as follows:

Husband’s age (years)	Wife’s age (years)	Husband’s age (years)	Wife’s age (years)
30	27	51	50
29	20	48	46
36	34	37	36
72	67	50	42
37	35	51	46
36	37	36	35

i) Draw a scatter plot of the data

ii) Write two important characteristics of the data that emerge from the scatter plot.

iii) Fit a linear regression model to the data and interpret the result in terms of the comparative change in the age of husband and wife.

iv) Calculate the standard error of regression and the coefficient of determination for the data.

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Question:

Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:

Customer Orders (in thousands)	Add-delete items (in thousands)	CPU time (in hours)
123.5	2.108	141.5
146.1	9.213	168.9
133.9	1.905	154.8
128.5	0.815	146.5
151.5	1.061	172.8
136.2	8.603	160.1
92.0	1.125	108.5

i) Find a linear regression equation that best fit the data.

ii) Estimate the error variance for the regression model obtained in i) above.

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Question:

Explain design and analysis of data of case control study in detail.

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Question:

Obtain sample size for the following given information:

$\delta =0.04,\, \pi _{1}=0.72,\, \pi _{2}=0.84,\alpha =0.02,\beta =0.25$

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Question:

Let P (w , w ) = 1 2 be a portfolio of two securities X and Y. Find the values of w1 and w2 in the following situations: i) ρxy = −1 and P is risk free. ii) σx = σy and variance P is minimum.

Variance P is minimum and ρxy = − ,5.0 σx = 2 and σy = 3

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Question:

Return distributions of the two securities are given below:

X	Y	p_xj= p_yj= p _j
0.16	0.14	0.33
0.12	0.08	0.25
0.08	0.05	0.17
0.11	0.09	0.25

Find which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

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Question:

Tumour is developing from the organ of a human body with concentration 9 2.3 ×10 with growth and decay control parameters 9.2 and 2.7 respectively. In how many days the size of the tumor will be twice?

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Question:

Let G )t( be the amount of the glucose in the bloodstream of a patient at time t. Assume that the glucose is infused into the bloodstream at a constant rate of k /g min . At the same time, the glucose is converted and removed from the bloodstream at a rate proportional to the amount of the glucose present. If at t = ,0 G = G )0( then

i) formulate the model.

ii) find )t(g at any time t.

iii) discuss the long term behavior of the model.

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Question:

If D+ and D ̶ denote presence and absence, respectively, of a disease and T+ and T ̶ denote test result as positive and negative, respectively, then on the basis of the following information:

	$D^{+}$	$D^{-}$	Total
$T^{+}$	145	2000	2145
$T^{-}$	15	48000	48015
Total	160	50000	50160

Find: (i) sensitivity (ii) specificity (iii) positive and negative predictive values.

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Question:

A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 5 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.

Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?