State whether the following statements are true or false and also give the reason in
support of your answer: (5×2=10)
(a)Vopt( _st )lies betweenV_prop andV (X_st ).and V_{Random (}X_st).
(b) The total number of all possible samples of size 3 without replacement from a population of size 7 is 21.
(c) While analysing the data of a 5 × 5 Latin Square design the d.f. for ESS is equal
to 16.
(d) In a Two-way Analysis of Variance test with 5 observations per cell having 4
blocks and 4 treatments the degree of freedom for the total variation is 64.
(e) The probability of selection of a sample of n from the population by SRSWOR is
1/ N.

Let's evaluate each statement one by one and determine whether they are true or false, along with the reasons for each answer:

(a) Vopt( \tilde{X}st ) lies between Vprop and V (Xst).and VRandom (Xst).

False. Vopt( \tilde{X}st ) refers to the optimal variance of the sample mean under some allocation method, which minimizes the variance. Vprop represents the variance of the sample mean under proportional allocation, and VRandom(Xst) represents the variance of the sample mean under simple random sampling without replacement (SRSWOR). It's important to note that Vopt( \tilde{X}st ) is always less than or equal to Vprop and VRandom(Xst), not necessarily between them. The reason for this is that the optimal allocation method aims to minimize the variance, so it will be less than or equal to the variances obtained under other allocation methods.

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