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Question:

In usual notations you are given the following information $\delta =0.05,\, \pi_1=0.75,\, \pi_2=0.65,\, \bar{\pi}=0.7,\alpha =0.05,\, \beta =0.20$ Find the sample size. If power of the test is 95% instead of 80%, then find new sample size.

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Question:

RTPCR test is applied on 300 covid patients and 200 non-covid patients. The results of the test are shown as follows.

		Disease status
		Yes (D ⁺ )	No (D ^- )	Total
Result of RTPCR test	_T+	480	40	520
	_T-	120	360	480
	Total	600	400	1000

What are the sensitivity and specificity of the test? Also, determine the positive and negative values of the test.

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Question:

What is the basic difference between (a) cross-sectional (b) cohort and (c) case control study designs. Explain with suitable examples of each design.

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Question:

A trial is conducted in which some people with disease X were randomly allocated into twogroups. First group was advised to do some morning walk for 30 minutes and take light foodeach day and second group was given one injection and one 100 mg tablet once a day tocontrol disease X. The injection can cause loose motion in some cases and 100 mg tablethas no side effect. At the end of two months, 90% of group I and 80% of group II had
recovered from disease X.
i) What are the regimens for group I and group II in this trial?
ii) What are the efficacies in group I and group II?
iii) What are the safety issues in group I and group II in this trial?

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Question:

If p denotes proportion and q denotes odds then prove that $q=\frac{p}{1-p}.$ Find the range of q. If the odds of smokers in a study are 0.25 then find the proportion of smokers in the study. Assume that each subject of the study is either a smoker or non-smoker.

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Question:

In the natural history of a disease define: total preclinical phase, detectable pre-clinical phase and clinical phase. Suppose on 10 am on 25.01.2010 a disease A onset you biologically. Suppose test of the disease A can detect it exactly after completion of 1000 days of biologically onset. Suppose signs and symptoms develop exactly after completion of 1010 days of biologically onset. Suppose you consult doctor exactly after 1015 days of biologically onset of the disease. Suppose outcome the treatment is cure. What is duration of (i) total preclinical phase (ii) detectable pre-clinical phase (iii) clinical phase.

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Question:

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

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Question:

Define the following:
(i) Covariance Matrix
(ii) Mahalanobis D2
(iii) Hotelling’s T2
(iv) Clustering
(v) Relationship between (ii) and (iii).

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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):

$x_1$	$x_2$	$x_3$	$x_4$	$y_1$	$y_2$	$y_3$	$y_4$
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	7.91
5	5	5	8	5.68	4.74	5.73

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

Let $X\sim N_4\left ( \mu ,\sum \right ),$ where $\mu =\begin{pmatrix} 3\\ -2 \\1 \\ -2 \end{pmatrix}$ and $\sum=\begin{pmatrix} 4 & 0 & 0 & 0\\0 & 3 & 0 &0 \\ 0 & 0 & 2 & -2\\ 0 & 0 & -2 & 5 \end{pmatrix}.$ Check the independence of the (i) $X_2$ and $X_1$ (ii) ( $X_2$ , $X_4$ ) and ( $X_1$ , $X_3$ ) (iii) ( $X_1$ , $X_2$ ) and ( $X_3$ , $X_4$ ).

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Question:

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

$\sum=\begin{pmatrix} 4 &-2 & 0\\ -2 & 4 &0 \\ 0 & 0 & 2 \end{pmatrix}.$ Determine the first principal component and the proportion of the total variability that it explains.

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Question:

$X=\binom{X^{(1)}}{X_{(2)}}\sim N_4\left ( \mu ,\sum \right ),$ where $\begin{pmatrix} -4\\ \frac{1}{4} \\ 0 \end{pmatrix}$ and $\sum=\begin{pmatrix} \begin{matrix} 2&2 \\ 2& 1\end{matrix} & \begin{matrix} 3&0 \\ 2& 0\end{matrix} \\ \begin{matrix} 3&2 \\ 0& 0\end{matrix} & \begin{matrix} 2&1 \\ 1& 1\end{matrix} \end{pmatrix}.$ Find the

$E(X^{(2)}|X^{(1)}=x^{(1)})$ and Cov $\left ( X^{(2)}|X^{(1)}=x^{(1)} \right ).$

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Question:

Let $\underset{\sim }{X}$ $=\binom{X_1}{X_2}$ has the following joint density function

$f\left ( X_1,X_2 \right )=\left\{\begin{matrix} 4x_1x_2 &,0<x_1<1,0<x_2<1, \\ 0 & ,\textup{otherwise.} \end{matrix}\right.$

Find the marginal distributions, mean vector and variance-covariance matrix. Also, comment on the independence of $X_1$ and $X_2$

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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 $\underset{\sim }{X}$ 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.

(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

Population mean when population variance is known and population is normal

Z-test

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

Population mean when population variance is unknown and population is normal

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

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

Population variance when the population is normal distributed

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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