IGNOU MST-004 Solved Assignment 2023 | PGDAST

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

a) If probability density function of a t-distribution is \(f(t)=\frac{1}{\pi\left(1+t^{2}\right)} ;-\infty<t<\infty\) then degrees of freedom of the distribution will be 1 .

b) If \(T_{1}\) and \(T_{2}\) are two estimators of an parameter \(\theta\) such that \(\operatorname{Var}\left(T_{1}\right)=1 / 2 n\) and \(\operatorname{Var}\left(T_{2}\right)=2 / n\) then \(T_{1}\) is more efficient than \(T_{2}\).

c) If the probability of non rejection of \(\mathrm{H}_{0}\) when \(\mathrm{H}_{1}\) is true is \(0.4\) then power of the test will be \(0.6\).

d) The Wilcoxon signed-rank test is more powerful than the sign test.

e) The t-test is used for testing the independence of two attributes.

2. a) A random sample of nine college students yielded the following data concerning the number of hours per day each student spent in using mobile phone:

\[
5,2,7,5.5,3.5,4,5,4.5,4
\]

Estimate the average number of hours per day spent in using mobile phone by the college students.

b) If the sample values are \(3,5,2,7\), and 0 then obtain the ML estimate for parameter \(\theta\) for the following distribution :

\[
\mathrm{f}(\mathrm{x}, \theta)=\theta \mathrm{e}^{-\theta \mathrm{x}} ; \quad 0 \leq \mathrm{x}, \theta>0
\]

3. A sample of 100 tyres is taken from a lot. The mean life of the tyres selected is the sample is found to be \(40,000 \mathrm{kms}\) with a standard deviation of \(3200 \mathrm{kms}\). Is it reasonable to suppose the mean life of tyres in the lot as \(41,000 \mathrm{kms}\) at \(5 \%\) level of significance? Also establish \(95 \%\) confidence limits within which the mean life of tyres in the lot is expected to lie.

4. The blood cholesterol levels of a population of workers have mean \(202 \mathrm{mg} / \mathrm{dl}\) and standard deviation \(14 \mathrm{mg} / \mathrm{dl}\). If a sample of 36 workers is selected from the population and sample mean is calculated then find

i) mean and standard error of the sampling distribution of the mean.

ii) approximate the probability that the sample mean of their blood cholesterol levels will lie between \(198 \mathrm{mg} / \mathrm{dl}\) and \(206 \mathrm{mg} / \mathrm{dl}\). 5. The following data relate to the number of items produced per shift followed normal distribution by two workers Rahul and Ramesh for a number of days:

Can it be inferred that Rahul is more stable worker compared to Ramesh by testing the variation in the item produced by them at \(5 \%\) level of significance.

6 a) In a city, 36 out of a random sample of 500 men were found to drinkers at a certain date. After the heavy increase in tax on intoxicants, another sample of 100 men in the same city included 6 drinkers. Do you feel that the observed proportion of drinkers decreasing significantly at \(1 \%\) level?

b) In a locality, 100 persons were randomly selected and asked about their educational achievements. The results were as follows:

Can we say that education depends on sex at \(5 \%\) level of significance?

7. Complete the following table:

8. A company is trying to improve the work efficiency of its employees. It has organized a special training programme for all employees. In order to assess the effectiveness of the training programme, the company has selected 10 employees randomly and administered a well-structured questionnaire. The scores (out of 100) obtained by the employees are given in the following table:

To examine whether the training programme has improved efficiency of the employees, give the answer of the following:

i) Are both samples are paired or independent?

ii) Formulate the null and alternative hypotheses.

iii) Which parametric test is used for testing the null hypothesis if it is known that the scores of the employees before and after the training programme follow the normal distribution? Conduct the test at \(1 \%\) level of significance and conclude the result.

iv) Which non-parametric test is used for testing the null hypothesis if it is known that the scores of the employees before and after the training programme do not follow the normal distribution but the distribution of the differences of scores before and after the training is symmetrical about its median? Conduct the test at \(1 \%\) level of significance and conclude the result.