Sample Solution

MST-005 Solved Assignment 2025

1. State whether the following statements are true or false and also give the reason in support of your answer:
   (a) The total number of all possible samples of size 2 without replacement from a population of size 7 is 21 .
   (b) Consecutive 3 random numbers starting from 8937 by ‘middle square method’ are 8937, 8699, 6726.
   (c) RBD is suitable in situations where it is not possible to divide the experimental material into a number of homogeneous blocks.
   (d) As we increase the sample size, representativeness of the population by the sample decreases.
   (e) In a big hall, there are 50 rows and each row has 60 students. A research scholar selects 10 rows randomly and then randomly selects 15 students from each selected row. It is an example of cluster sampling procedure.
2. (a) Draw all possible samples of size 2 from the population [2, 3, 4] and verify that $E(\overline{x})=\overline{X}$$E(\overline{x})=\overline{X}$E( bar(x))= bar(X)E(\bar{x})=\bar{X}$E(\overline{x})=\overline{X}$. Also find variance of $\overline{x}$$\overline{x}$bar(x)\bar{x}$\overline{x}$.
   (b) A sample of 60 students is to be drawn from a population consisting of 600 students belonging to two villages, A and B . The means and standard deviations of their marks are give below:

3. To determine the yield rate of wheat in a district of Punjab, 6 groups of 6 plots each were constructed. The data are given in the following table:

6. In the following data, two values are missing. Estimate these values by Yates method and analyse the data by suitable technique.

7. Identify the design given in the following table and then carry out the analysis:

8. (a) The distribution function of Pareto distribution is given by

Solution

State whether the following statements are true or false and also give the reason in support of your answer:

Answer:

• True.
• Justification: The number of ways to choose 2 items from 7 without replacement is given by the combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$$C(n,k)=\frac{n!}{k!(n-k)!}$C(n,k)=(n!)/(k!(n-k)!)C(n, k) = \frac{n!}{k!(n-k)!}$C(n,k)=\frac{n!}{k!(n-k)!}$. Here, $n=7$$n=7$n=7n = 7$n=7$, $k=2$$k=2$k=2k = 2$k=2$, so $C(7,2)=\frac{7\cdot 6}{2\cdot 1}=21$$C(7,2)=\frac{7\cdot 6}{2\cdot 1}=21$C(7,2)=(7*6)/(2*1)=21C(7, 2) = \frac{7 \cdot 6}{2 \cdot 1} = 21$C(7,2)=\frac{7\cdot 6}{2\cdot 1}=21$. Thus, there are 21 possible samples.

• False.
• Justification: In the middle square method, a number is squared, and the middle digits are extracted to form the next number. Start with 8937:
  • ${8937}^2=79,869,969$${8937}^2=79,869,969$8937^(2)=79,869,9698937^2 = 79,869,969${8937}^2=79,869,969$. For a 4-digit number, extract the middle 4 digits (positions 3–6, assuming 8 digits): 9869.
  • Next, ${9869}^2=97,397,161$${9869}^2=97,397,161$9869^(2)=97,397,1619869^2 = 97,397,161${9869}^2=97,397,161$. Middle 4 digits: 7397.
  • The sequence is 8937, 9869, 7397, not 8937, 8699, 6726. Thus, the given sequence is incorrect.

• False.
• Justification: Randomized Block Design (RBD) requires dividing experimental material into homogeneous blocks to control for variability within blocks. If it’s not possible to form homogeneous blocks, RBD is unsuitable, and other designs (e.g., completely randomized design) may be considered.

• False.
• Justification: Larger sample sizes generally increase representativeness, as they reduce sampling error and better capture the population’s characteristics, assuming random sampling. For example, in a population with mean $\mu$$\mu$mu\mu$\mu$, the sample mean’s variance decreases as sample size $n$$n$nn$n$ increases (by the formula $\text{Var}(\overline{x})=\frac{{\sigma }^2}{n}$$\text{Var}(\overline{x})=\frac{{\sigma }^2}{n}$“Var”( bar(x))=(sigma^(2))/(n)\text{Var}(\bar{x}) = \frac{\sigma^2}{n}$\text{Var}(\overline{x})=\frac{{\sigma }^2}{n}$).

• False.
• Justification: This is two-stage sampling, not cluster sampling. In cluster sampling, entire clusters (e.g., rows) are randomly selected, and all units within selected clusters are sampled. Here, only 15 students are randomly selected from each of the 10 chosen rows, not all 60. This resembles a combination of cluster and simple random sampling within clusters.