State whether the following statements are true or false and also give the reason in support of your answer.
c) In SRSWOR, the possible numbers of sample of size n from a population of size N if sampling is done with replacement is Nⁿ
.d) One-way analysis of variance is a generalization of the two sample t-test.
e) If experimental error is reduced considerably and the efficiency of the design is
decreased.
f) If strata are heterogeneous then stratified sampling schemes provides estimates with greater precision.
g) If one wants to convert random numbers selected from two digit numbers 00-99 to
uniformly distributed U (0, 1) variables then one has to divide them by 99.

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.

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. 

. Complete the following table:

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?

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?

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.

The blood cholesterol levels of a population of workers have mean 202 mg/dl and standard deviation 14 mg/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 mg/dl and 206 mg/dl. 

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 kms with a standard deviation of 3200 kms. Is it reasonable to suppose the mean life of tyres in the lot as 41,000 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.

If the sample values are 3, 5, 2, 7, and 0 then obtain the ML estimate for parameter θ for the following distribution :

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.

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

then degrees of freedom of the distribution will be 1.
b) If T1 and T2 are two estimators of an parameter θ such that Var(T1) = 1/2n and
Var(T2) = 2/n then T1 is more efficient than T2.
c) If the probability of non rejection of H0 when H1 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.

Events A, B, C are mutually exclusive and exhaustive. If odds against A are 4:1 and against B are 3: 2 . Find the odds against event C.

A person is known to hit the target in 3 out of 4 shots whereas another person is known to hit 2 out of 5 shots. Find the probability that the target being hit when they both try.

A die is rolled. If the outcome is a number greater than 2, what is the probability that it is an odd prime number?

If the probability that an individual suffers a bad reaction from an injection of a given
 serum is 0.002, determine the probability that out of 400 individuals
(i) exactly 2
(ii) more than 3
(iii) at least one
 individuals suffer from bad reaction. 

Comment on the statement: “The mean of a binomial distribution is 4 and variance 5”.

Let X and Y be two independent random variables such that  and . Find 

A player tosses two unbiased coins. He wins Rs. 10 if 2 heads appear, Rs. 5 if one head appears and Rs 1 if no head appears. Find the expected value of the amount won by him

A rain coat dealer can earn Rs 800 per day during a rainy day. If it is a dry day, he can lose Rs 150 per day. What is his expectation, if the probability of rain is 0.6?