Solve your IGNOU Doubts

Question:

A factory purchases bolts in lots of 800. Acceptance is decided using a single-sampling plan with sample size n = 20 and acceptance number c = 3. Assume that 2% defective items are considered acceptable quality and 7% defective items are considered unacceptable quality.

Find:

(i) The probability of accepting a lot when the incoming quality level is 5% defective.

(ii) The Average Outgoing Quality (AOQ), assuming rejected lots are completely screened and defectives are replaced.

(iii) The Average Total Inspection (ATI).

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Question:

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

(i) The c-chart is suitable for moniterting to proportion of defective.

(ii) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.

(iii) Autocorrelation measures the relationship between yt and yt-k after removing the effect of intermediate lags.

(iv) If the effect of summer and winter is not constant on the sale of AC then we use the additive model of the time series.

(v) In a series system, improving the reliability of the weakest component gives the maximum improvement in system reliability.

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Question:

Describe the following:

(i) t-distribution

(ii) General procedure of Testing of hypothesis

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Question:

Let X1, X2, ..., Xbe a random sample from the Exponential distribution with pdf f(x; 0) = 0e-ex, x > 0, 0 > 0

(i) Obtain the maximum likelihood estimator (MLE) of 0.

(ii) Hence, find the maximum likelihood estimate of for the observed sample: 2.1, 1.6, 3.4, 0.9, 2.5

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Question:

A researcher wants to compare the average test scores of students taught using Method A and Method B. A random sample of students was selected from each group, and their scores are given below:

Student	Method A	Method B
1	68	72
2	74	75
3	71	78
4	69	70
5	73	76
6	70	74
7	72	77
8	75	79

Assuming that the populations are normally distributed with equal variances. For testing whether there is a significant difference between the mean scores of the two teaching methods at the 5% level of significance

(i) State the null and alternative hypotheses.

(ii) Name the appropriate t-test and justify your choice.

(iii) Compute the test statistic and critical value.

(iv) State your conclusion.

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Question:

A survey was conducted to study the preference for online learning among students from urban and rural areas. Out of 300 urban students, 210 preferred online learning, while out of 250 rural students, 150 preferred online learning. Construct a 95% confidence interval for the difference between the population proportions of students preferring online learning in urban and rural areas.

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Question:

Let $X_1, X_2, \dots, X_n$ be a random sample from a population with mean $\mu$ and variance $\sigma^2$ . Consider the estimator
$$\hat{\mu} = \frac{1}{n} \sum_{i=1}^{n} X_i + 2$ $
(i) Find the bias of the estimator $\hat{\mu}$ .
(ii) Find the variance of $\hat{\mu}$ .
(iii) Hence, calculate the Mean Square Error (MSE) of $\hat{\mu}$ .

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Question:

A school caretaker looks after 6 children. The ages (in years) of the children are given below:

Child	Age (in years)
Meena	5
Aarav	4
Nisha	6
Kabir	4
Pooja	5

(i) What is the nature (type) of the population of the ages of children?
(ii) Construct the sampling distribution of the sample mean for all possible samples of size n = 2.
(iii) Comment on whether the sampling distribution is approximately normal.
(iv) Find the mean and standard error of the sampling distribution. (20)

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Question:

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

(i) The mean square error is calculated when the estimator is unbiased.

(ii) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.

(iii) In the method of moments, population moments are equated to corresponding sample moments.

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

(v) The chi-square distribution is symmetric for all degrees of freedom.

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Question:

Define clustering. Differentiate between single linkage and complete linkage method of clustering.
(b) Consider the following partitioned sample variance-covariance matrix S, obtained from a sample of size 10.
$$S = \left( \begin{array}{cc|cc} 4.8 & 2.9 & 0.2 & -0.2 \\ 2.9 & 7.9 & 0.0 & 0.7 \\ \hline 0.2 & 0.0 & 7.1 & 7.2 \\ -0.2 & 0.7 & 7.2 & 11.3 \end{array} \right)$ $
If $|S| = 799.13$ , then test for the independence of the sub-vectors $X^{(1)}_{2 \times 1}$ and $X^{(2)}_{2 \times 1}$ .
$\quad$ (Use $\Lambda_{2, 2, 7}(0.05) = 0.230$ )
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Question:

If $\mathbf{X} \sim N_3 (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$ and $\boldsymbol{\Sigma} = \begin{pmatrix} 5 & 3 & 0 \\ 3 & 3 & -2 \\ 0 & -2 & 5 \end{pmatrix}$ . Then find the joint distribution
of X₁ + 2X₂, 2X₁ - X₂ and X₃.

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Question:

Obtain the square root matrix corresponding to a matrix $\mathbf{A} = \begin{pmatrix} 3 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & 1 & 3 \end{pmatrix}$ . Also
verify that $\mathbf{A}^{1/2} \mathbf{A}^{1/2} = \mathbf{A}$ .

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Question:

Let $\mathbf{X} \sim N_4 (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ , where $\boldsymbol{\mu} = \begin{pmatrix} 4 \\ -1 \\ 6 \\ -1 \end{pmatrix}$ and $\boldsymbol{\Sigma} = \begin{pmatrix} 5 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & -1 & 1 \end{pmatrix}$ . Check the independence
of (i) X₂ and X₁ (ii) (X₂, X₄) and (X₁, X₃) (iii) (X₁, X₂) and (X₃, X₄).

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Question:

Let two independent samples of size $\text{N}_{1} = 10$ and $\text{N}_{2} = 12$ from trivariate normal populations have following mean vectors and variance-covariance matrices:
$\bar{\text{x}}_{1} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ and $\text{S}_{1} = \begin{pmatrix} 1.2 & 0.3 & 0.2 \\ 0.3 & 1.5 & 0.4 \\ 0.2 & 0.4 & 1.8 \end{pmatrix}$
$\bar{\text{x}}_{2} = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}$ and $\text{S}_{2} = \begin{pmatrix} 1.0 & 0.2 & 0.3 \\ 0.2 & 1.4 & 0.3 \\ 0.3 & 0.3 & 1.6 \end{pmatrix}$ , respectively.
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$$If \mathbf{S}_p^{-1} = \begin{pmatrix} 1.138 & -0.152 & -0.181 \\ -0.152 & 0.837 & -0.125 \\ -0.181 & -0.125 & 0.740 \end{pmatrix} \text{is the inverse of the pooled covariance matrix}$ $
$$\mathbf{S}_p = \begin{pmatrix} 1.09 & 0.245 & 0.255 \\ 0.245 & 1.445 & 0.345 \\ 0.255 & 0.345 & 1.69 \end{pmatrix}, \text{then test whether the mean vectors of the two}$ $
samples are significantly different from each other at 5% level of significance or not.
$$\text{(Use } F_{3, 18}(0.05) = 3.16 \text{ )}$ $

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Question:

Let $\text{X} = \begin{pmatrix} \text{X}^{(1)} \\ \text{X}^{(2)} \end{pmatrix} \sim \text{N}_{4} (\mu, \Sigma)$ , where $\underset{\sim}{\mu} = \begin{pmatrix} -4 \\ 1 \\ \hline 4 \\ 0 \end{pmatrix}$ and $\Sigma = \left( \begin{array}{cc|cc} 2 & 0 & 1 & 0 \\ 0 & 2 & 2 & 0 \\ \hline 1 & 2 & 6 & 1 \\ 0 & 0 & 1 & 1 \end{array} \right)$ .
Then find $\text{E}(\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)})$ and $\text{Cov}(\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)})$ .

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Question:

Let $\underset{\sim}{\text{X}}_{p \times 1}$ , $\underset{\sim}{\mu}_{p \times 1}$ and $\Sigma_{p \times p}$ be partitioned as $\underset{\sim}{\text{X}}_{p \times 1} = \begin{pmatrix} \text{X}_{k \times 1}^{(1)} \\ \text{X}_{(p-k) \times 1}^{(2)} \end{pmatrix}$ , $\underset{\sim}{\mu}_{p \times 1} = \begin{pmatrix} \mu_{k \times 1}^{(1)} \\ \mu_{(p-k) \times 1}^{(2)} \end{pmatrix}$ and
$\Sigma_{p \times p} = \begin{pmatrix} \Sigma_{11 \ k \times k} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$ . Then derive the conditional distribution of $\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)}$ .

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Question:

State whether the following statements are True or False. Give reasons in support of your answer: 5×2=10
(a) The eigenvalues of a positive definite matrix are less than zero.
(b) $A = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}$ is an orthogonal and idempotent matrix.
(c) A factor model postulates that a random vector $\underline{X}$ is linearly dependent upon a few observable common factors.
(d) A given px1 vector $\underline{\mu}_0$ is a plausible value for the mean of a multivariate normal distribution can be tested using Student's t test.
(e) In single linkage method, we find the minimum distance in the distance matrix and merge the minimum distanced clusters